Acta hydrotechnica 31/54 (2G18), Ljubljana Open Access Journal Odprtodostopna revija ISSN 1581-0267 UDK/UDC: 532.517:621.6 Prejeto/Received: 17.09.2018 Sprejeto/Accepted: 30.11.2018 Kratki znanstveni članek - Short scientific paper Simulation of fluid sloshing in a tank using the SPH and Pendulum METHODS Simulacija pljuskanja kapljevine v cisterni po metodi hidrodinamike ZGLAJENIH DELCEV (SPH) IN PO METODAH NIHALA Vesna Vidmar1, Gregor Pethovšek2, Dušan Žagar1'* 1 Fakulteta za gradbeništvo in geodezijo, Univerza v Ljubljani, Jamova cesta 2, 1000 Ljubljana, Slovenija 2 HR Wallingford, Howbery Park, Wallingford, Oxfordshire OX10 8BA, United Kingdom Abstract We simulated liquid sloshing in a circular road tanker during two typical manoeuvres, namely steady-turn and lane change. A quasi-static pendulum, a modified dynamic pendulum with adjustable rod length, and the SPH (Smooth Particle Hydrodynamics) model Tis Isat were applied to simulate liquid oscillations. A simplified vehicle-tanker overturning model was developed and applied in order to determine the overturning threshold for the first manoeuvre. The agreement of liquid oscillations between the applied methods was better in the second manoeuvre, while the maximum inclination of the liquid gravity centre was successfully simulated in both cases. Both dynamic methods, the SPH and the dynamic pendulum, show a significantly lower overturning threshold, while all methods show similar overturning behaviour for a vehicle with liquid cargo. The threshold computed using the SPH model is slightly higher than with the dynamic pendulum due to wall-particle and particle-particle interactions. The results and comparisons confirm the suitability of the SPH method for simulating sloshing in road tankers and also show the method's advantages: realistic description of the non-linear free surface in real-time, along with a consideration of mixing and friction processes within the fluid. Keywords: SPH method, pendulum, fluid oscillations, closed domain, liquid cargo, overturning threshold. V prispevku obravnavamo pljuskanja kapljevine v cisterni krožnega prečnega prereza med dvema tipičnima manevroma: vstopom in vožnjo skozi krožišče ter menjavo voznega pasu. Za simulacije nihanja kapljevine smo uporabili tri metode: modela kvazistatičnega nihala in modificiranega dinamičnega nihala s spremenljivo ročico ter model Tis-Isat, ki deluje po metodi hidromehanike zglajenih delcev (SPH - Smooth Particle Hydrodynamics). Za določitev praga prevrnitve pri prvem manevru smo izdelali in uporabili model poenostavljenega vozila s cisterno. Ujemanje nihanja kapljevine je bilo boljše pri drugem manevru, maksimalni odmik težišča kapljevine pa smo uspešno simulirali v obeh obravnavanih primerih. Vse tri metode pokažejo podobno obnašanje vozila s tekočim tovorom ob prevrnitvi, obe dinamični metodi, SPH in dinamično * Stik / Correspondence: dusan.zagar@fgg.uni-lj.si © Vidmar V. et al.; Vsebina tega članka se sme uporabljati v skladu s pogoji licence Creative Commons Priznanje avtorstva -Nekomercialno - Deljenje pod enakimi pogoji 4.0. © Vidmar. et al.; This is an open-access article distributed under the terms of the Creative Commons Attribution - Non Commercial -ShareAlike 4.0 Licence. https://doi.org/10.15292/acta.hydro.2018.04 Izvleček 51 Vidmar V. et al.: Simulation of fluid sloshing in a tank using the SPH and Pendulum methods - Simulacija pljuskanja kapljevine v cisterni po metodi hidrodinamike zglajenih delcev (SPH) in po metodah nihala Acta hydrotechnica 31/54 (2018), 51-65, Ljubljana nihalo, pa dajeta znatno nižji prag prevrnitve. Pri SPH je prag prevrnitve nekoliko višji kot pri dinamičnem nihalu zaradi interakcije med delci kapljevine ter kapljevino in ostenjem. Primerjava rezultatov dinamičnih metod potrjuje uporabnost metode SPH za simulacije pljuskanja v tovornih cisternah, prav tako pa tudi prednosti metode SPH: opis nelinearne proste gladine v realnem času ter upoštevanje viskoznih procesov in mešanja v kapljevini. Ključne besede: metoda SPH, nihalo, nihanje kapljevine, zaprto območje, tekoči tovor, prag prevrnitve. 1. Introduction Traffic accidents with a rollover of liquid cargo are relatively frequent and the environmental effect of spilled fluids can be severe, as the fluids being transported often contain toxic and environmentally hazardous substances. The exact mechanism of overturning is not yet well understood. Various methods have been applied to describe the forces of sloshing liquid on the tank and the vehicle. Since the 1960s various pendulum-based methods have been used for calculations with several tank shapes (Rattaya, 1965) and in developing simplified models of liquid sloshing (Abramson, 1966; Rattaya, 1965). Computations are usually based on the analogy between the movement of the liquid gravity centre and the oscillations of a pendulum (Aliabadi et al., 2003; Casasanta, 2010; Salem, 2000). The pendulum's parameters are determined by the shape of the tank and the type of liquid (Rattaya, 1965). Several studies on two- and three-dimensional sloshing were also carried out using finite element methods (Aliabadi et al., 2003; Behr and Tezduyar, 1994; Okamoto and Kawahara, 1990; Ramaswamy et al., 1986; Wu et al., 1998). Furthermore, analytical solutions were explored (Kolaei et al., 2014) as well as other numerical methods, such as boundary element methods (Kolaei et al., 2015) and the particle based MSP (moving particle semi-implicit) method (Jena and Biswal, 2017). Recently the SPH (smoothed particle hydrodynamics) method has been applied for research studies on similar phenomena due to its ability to simulating the exact free surface, the interactions between the particles, and the interactions between the liquid and the solid boundaries. The SPH method's appropriateness for simulating sloshing in tankers and tanks has been examined and verified against laboratory experiments and comparisons. Various SPH models have been used recently to perform such simulations (Cao et al., 2014; Shao et al., 2012; Vorobyev et al., 2011). In this study we applied the Tis Isat model (Dzebo et al., 2014, 2013; Petkovsek et al., 2010) developed at the Faculty of Civil and Geodetic Engineering of the University of Ljubljana. Assessing the forces of sloshing liquid facilitates computation of tanker's (vehicle's) overturning. The threshold of vehicle overturning was studied extensively by Issermann (1976), who also proposed an analytical model. Another analytical model was proposed by Kolaei, Rakheja, and Richard (Kolaei et al., 2014). Strandberg (1978) compared the overturning of vehicles carrying solid and liquid cargo based on laboratory experiments and examined the dynamic forces caused by splashing during the overturning of a vehicle. Rakheja and Ranganathan (1993) proposed a simple methodology to estimate the rollover threshold of partially filled liquid cargo vehicles. Salem (2000) developed 2D and 3D models using the pendulum method. Kang et al. (2001) analysed partially filled tanks of various cross sections (circular, modified oval and two optimized cross sections). They also performed a sensitivity analysis for several parameters describing the roll stability characteristics of vehicles carrying partially filled tanks. Modaressi-Tehrani et al. (2006) analysed the overturning moment caused by transient liquid slosh inside a partially-filled moving tank with different overturning models. In this study we present a modification of the method described in Aliabadi et al. (2003) by using a pendulum rod of adjustable length. We furthermore present a comparison of forces during a typical manoeuvre computed with the SPH model Tis-Isat and the modified pendulum model. The forces and the moments due to sloshing are applied in a simplified overturning model. The physical 52 Vidmar V. et al.: Simulation of fluid sloshing in a tank using the SPH and Pendulum methods - Simulacija pljuskanja kapljevine v cisterni po metodi hidrodinamike zglajenih delcev (SPH) in po metodah nihala Acta hydrotechnica 31/54 (2018), 52-65, Ljubljana principles behind the oscillations of a solid body (pendulum) and particle-described liquid (SPH) differ significantly; we therefore expect significant differences in forces, moments, and overturning threshold between the two methods. We also expect to obtain a more detailed description of liquid behaviour using the SPH method. The main aim of the performed study was to estimate the rollover threshold of the vehicle and to compare the differences between the two applied methods with regard to the computation of forces and moments in partially filled tanks with increasing free surface levels. 2. Methods and materials 2.1 SPH model Tis-Isat Tis-Isat is a mesh-free, particle-based numerical model based on the SPH method (Gingold and Monaghan, 1977; Lucy, 1977). It solves the mass conservation equation, momentum equation, and the equation of state for weakly compressible fluids, and also introduces a non-discrete boundary condition with friction into the SPH method (Petkovsek et al., 2010). The derivation of basic equations is described in numerous sources (Delorme, 2009; Gomez-Gesteira et al., 2010; Jones and Belton, 2006; Liu and Liu, 2003; Monaghan, 1992). In the Tis Isat model we used the cubic spline kernel function proposed by Monaghan and Lattanzio (1985), the Verlet algorithm for time stepping (Verlet, 1967), and additional density flux between particles (Molteni and Colagrossi, 2008) for preventing oscillations of the pressure field. Reinitialisation was performed every 10 time steps in order to prevent the simulations from instability. The Tis-isat model was verified against two benchmark experiments. The modelling of two-dimensional water column collapse (Martin and Moyce, 1952) and the propagation of dam break wave (Rajar, 1972) are described in Petkovsek et al. (2010) and Dzebo et al. (2014), respectively. 2.2 Pendulum model Several pendulum methods are available for simulating fluid sloshing in circular tanks (Abramson, 1966; Aliabadi et al., 2003; Casasanta, 2010; Ranganathan et al., 1993; Rattaya, 1965; Salem, 2000). We modified the pendulum method for circular tanks described in Aliabadi et al. (2003) by using a pendulum rod of adjustable length. The following assumptions are valid for the pendulum (Figure 1): (i) the mass of liquid is concentrated in the gravitational centre of the fluid at the end point of the rod; (ii) the rod is rotating around a frictionless pivot in the fixed centre point of the tank S; (iii) a constant inertial force is impacting the pendulum due to centrifugal acceleration. The force in the rod Fv is computed from the oscillation equations and depends on the parameters of the model: arm length l and fluid mass mL. These depend on the level of fluid in the tank and the fluid density. Following Aliabadi et al. (2003), the governing equation of pendulum motion due to gravitational (g) and radial (qy) accelerations is: ' (1) (2) 9 = j (aY • cos 9 — g • sin 9) a — dl — d-1 d t = dt2 where 9 denotes the angle between the rod and the vertical axis, while the two derivatives over time (9 and 9) denote angular velocity and angular acceleration. 9 and 9 can be computed from the acceleration using trapezoidal numerical integration in each time step with the following initial conditions: 0|t=o = 0, = 0. The forces Fv computed using the modified model were compared to the results of the original model (Aliabadi et al., 2003). We obtained excellent agreement for four levels of liquid using a standard calibration manoeuvre, where the horizontal accelerations qy arise due to the tanker traveling at a constant speed of 10 ms-1 through a curve with a constant radius 250 m (Vidmar, 2012), Figures 6 and 7 therein). 52 Vidmar V. et al.: Simulation of fluid sloshing in a tank using the SPH and Pendulum methods - Simulacija pljuskanja kapljevine v cisterni po metodi hidrodinamike zglajenih delcev (SPH) in po metodah nihala Acta hydrotechnica 31/54 (2018), 53-65, Ljubljana 2.3 Overturning models 2.3.1 Quasi-static pendulum overturning model In the first approximation of rollover threshold computations we used a two-dimensional simplified overturning model presented by Rakheja and Ranganathan (1993). We assume that the forces acting on vehicle do not change temporally and refer to the model as quasi-static. If we assume (i) a small roll angle; (ii) a low unsprung weight; (iii) that the roll centre is in the ground plane, then we can determine the point of roll instability. Vehicle rollover occurs when the overturning moment exceeds the restoring moment; then, the horizontal acceleration is calculated as (Rakheja and Ranganathan, 1993): (WT+WL) WThT+WL(hL+R-Ztt) ■T (3) where aYn denotes normalized lateral acceleration with respect to g, WT is the vehicle weight without cargo, WL is the weight of liquid cargo, hT and hL denote the central gravity heights of the vehicle and the cargo, respectively, R is the diameter of the tank, T is the vehicle's half-width, and Ztt is the initial distance between the bottom of the tank and the gravity centre of the fluid, depending on the quantity (level) of fluid in the tank (Figures 1 and 2). 2.3.2 Dynamic pendulum overturning moment For the final computations of rollover threshold and comparison to the SPH model, we used dynamic equations in the pendulum model. The acceleration of the pendulum aYp due to the radial acceleration (ay) and the gravity (g) is: aYp — —g • sin 8 + aY • cos 8 (4) while the force in the rod Fv and the force of liquid on the tank FC can be written as: Fv — (—g • cos 8 + aYp • sin 8) • mL Fc — —Fv, (5) (6) where mL denotes the mass of the liquid. The two terms in the overturning moment (MO) equation M0=MT + ML (7) denote the contributions of the vehicle (Mr) and the oscillating liquid (Ml). These are further calculated as: Mt — FT-hT Ft — —mT • aY , (8) (9) Ml — Fc-Hs where Ft denotes the force of the vehicle, with the vehicle's mass denoted by mr, and (10) where Hs is the lever in the oblique plane: Hs = hs^sm8 (11) and hs is the distance between the centre of the tank and ground. 2.3.3 SPH method overturning moment In the SPH model (Figure 3), the equations for calculating the overturning moment are as follows: = (12) which is an equivalent to Eq. (10) in the j-th time step, where Mcij — Fcij x djj (13) and FCij is the force of the 7-th particle on the tank in j-th time step. Here FCij — — FDij (14) and FDij can be written as Foij = \_pYDij FzDij\ (15) where FYDij and FZDij are represented by: PvDij = mi • anj (16) PzDij = mi • azij (17) and m,i and are the mass and the position vector of the 7-th particle, respectively. The last vector in Eq. (13) is: djj — [dYij dzij\ (18) Acceleration of the 7-th particle in the j-th time step is further calculated from particle velocities as: aYij — aY + azij — -g + vYlj(t+dt)-vYlj(t) dt vZij(t+dt)-vZij(t) dt (19) (20) 52 Vidmar V. et al.: Simulation of fluid sloshing in a tank using the SPH and Pendulum methods - Simulacija pljuskanja kapljevine v cisterni po metodi hidrodinamike zglajenih delcev (SPH) in po metodah nihala Acta hydrotechnica 31/54 (2018), 54-65, Ljubljana Figure 1: Pendulum model. Slika 1: Model nihala. Figure 2: Overturning mechanism in the pendulum method. Slika 2: Prevrnitveni mehanizem pri metodi nihala. 52 Vidmar V. et al.: Simulation of fluid sloshing in a tank using the SPH and Pendulum methods - Simulacija pljuskanja kapljevine v cisterni po metodi hidrodinamike zglajenih delcev (SPH) in po metodah nihala Acta hydrotechnica 31/54 (2018), 55-65, Ljubljana V Fo{ 1f ' Fgt i h i i t FZi <-2T Figure 3: Overturning mechanism in the SPH method. Slika 3: Prevrnitveni mehanizem pri metodi SPH. Table 1: Vehicle and cargo parameters. Table 2: Pendulum and SPH parameters. Preglednica 1: Parametri vozila in tovora. Preglednica 2: Parametri nihala in metode SPH. Truck Filling [%] Wl [N] Particles Weight 46882 N 10 20511 640 Trailer and cargo 20 56113 1752 Weight (empty) 87448 N 30 99442 3104 Diameter 2.00 m 40 147214 4595 Centre height of trailer and tank 2.29 m 50 197058 6151 Center height of tank 3.29 m 60 246902 7707 70 294675 9198 80 338003 10551 90 373606 11662 FZ2 > 52 Vidmar V. et al.: Simulation of fluid sloshing in a tank using the SPH and Pendulum methods - Simulacija pljuskanja kapljevine v cisterni po metodi hidrodinamike zglajenih delcev (SPH) in po metodah nihala Acta hydrotechnica 31/54 (2018), 56-65, Ljubljana Several time-step lengths were tested in order to achieve stable simulations and a smooth graphic representation of results. We adopted dt = 10-4 s in all SPH simulations. The restoring moment of the observed system depends on the gravity centre position of the sloshing liquid; when depicted by a pendulum, the gravity centre can be determined from the pendulum's inclination, while when using the SPH method it can be calculated from the particles' position vectors. In order to be consistent with the quasi-static pendulum we adopted another simplification suggested by Rakheja and Ranganathan (1993), and Ranganathan et al. (1993). A small inclination was adopted for the pendulum in Eq. (3), inducing the maximum restoring moment (of static cargo) to resist to the overturning. Even though the inclination can be significant, particularly with lower liquid levels, we adopted the small angle principle in order to enable a comparison of the three applied methods. The resisting moment is therefore independent of the applied method: Mr = (mT + mL) • g •T (21) where mT and mL are the mass of the vehicle and the liquid cargo, respectively; g is the gravitational constant and T is the half width of the vehicle (Figure 2). In all performed computations we used a vehicle described by the parameters in Table 1, while the pendulum and SPH parameters are described in Table 2. 2.4 Manoeuvres We present two characteristic manoeuvres: (i) lateral liquid cargo shifting during a steady turn -driving through a curve with constant radius (Rakheja and Ranganathan, 1993); (ii) sloshing due to a lane change manoeuvre - avoiding an obstacle by changing the lane, a standard NATO AVTP 03-160W test (Casasanta, 2010). The input data for each manoeuvre are presented below. 2.4.1 Steady turn In this manoeuvre the vehicle turns instantaneously from a straight section into a constant radius curve at time ts. In both pendulum methods the radial acceleration applied as the input data was: ( 0,0