Journal of Fluids Engineering. Received November 23, 2016; Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242 Copyright (c) 2017 by ASME ed ite d Investigation of Wave Characteristics in Oscillatory Motion of Partially Filled Rectangular M. Ozbulut∗ ot Co py Tanks tN Piri Reis University sc rip Engineering Faculty Istanbul, Turkey Ma nu Email: [email protected] N. Tofighi University of Victoria Ac ce pt ed Department of Mechanical Engineering Victoria, Canada Email: [email protected] O. Goren Istanbul Tecnical University Faculty of Naval Architecture and Ocean Engineering Istanbul, Turkey Email: [email protected] Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use Journal of Fluids Engineering. Received November 23, 2016; Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242 Copyright (c) 2017 by ASME M. Yildiz Integrated Manufacturing Technologies Research and Application Center, Sabanci University, Tuzla, 34956, Istanbul, Turkey. Composite Technologies Center of Excellence, ed ite d Sabanci University-Kordsa, Istanbul Technology Development Zone, py Sanayi Mah. Teknopark Blvd. No: 1/1B,Pendik, 34906 Istanbul, Turkey. Co Faculty of Engineering and Natural Sciences, Sabanci University, Tuzla, 34956 Istanbul, Turkey. sc rip tN ot Email: [email protected] ABSTRACT Ma nu Simulations of oscillatory motion in partially filled rectangular tanks with different tank geometries, fullness ratios and motion frequencies are presented. Smoothed Particle Hydrodynamics (SPH) method is used to discretize the governing equations together with new velocity ed variance based free surface and artificial particle displacement algorithms to enhance the robustness and the accuracy of the numerical scheme. 2-D oscillatory motion is investigated for pt three different scenarios where the first one scrutinizes the kinematic characteristics in reso- ce nance conditions, the second one covers a wave response analysis in a wide range of enforced Ac motion frequencies, and the last one examines the dynamic properties of the fluid motion in detail. The simulations are carried on for at least 50 periods in the wave response analysis. It is shown that numerical results of the proposed SPH scheme are in match with experimental and numerical findings of the literature. ∗ Corresponding author. Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use Journal of Fluids Engineering. Received November 23, 2016; Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242 Copyright (c) by ASME 1 2017 INTRODUCTION The inner free-surface flows in partially filled tanks have significant effects on the general stability and motion of the ships having large liquid cargo tanks such as LNG ships [1, 2]. The solution of coupled sloshing-seakeeping problems posses a great importance on the preliminary and concept design processes of ships [3] and new researches on this issue are still needed [4]. Depending on the enforced ed ite d frequency of the motion and fullness ratio of the tank, the fluid motion inside the tank may become extremely complex and hence requires long term analysis to understand its characteristics. If the tank is forced to surge around the lowest natural frequency, the resonant fluid motion inside the tank becomes py important. Therefore, enforced surge or pitch motion of liquids in a tank has attracted the attention of Co engineers and scientists in the field of hydrodynamics. To this end, one may find several theoretical, experimental and numerical studies in the literature that have tried to explore the complicated physics ot behind the oscillatory motion inside a partially filled tank [5–10]. tN Faltinsen and Timokha [11] modeled the surge and pitch excited resonant sloshing motion by adaptive multi-modal theory, assuming that the flow is irrotational and has no overturning waves. They stated sc rip that direct numerical methods (finite difference, finite element and boundary element methods) experience some difficulties in volume and energy conservation as well as in accurate description of fluid Ma nu impact on the tank walls during long term simulations. Their reported results are in a good agreement with the experimental observations provided that the ratio of water depth to the tank beam is larger than 0.24. To further extend the scope of the theory, Faltinsen and Timokha [12] proposed an asymptotic model approximation to examine nonlinear resonant waves for depth/breadth ratios between 0.1 and ed 0.24. In addition to theoretical models, numerous studies involving a variety of numerical methods such pt as marker and cell (MAC), volume of fluid (VOF), level set (LS) and hybrid VOF-LS can be found in ce literature. Chen et al. [13] gave a list of numerical studies dedicated to excited sloshing motion of liquids Ac and discussed advantages and disadvantages of these schemes. They stated that the inability to capture the topology of the free-surface region and low numerical accuracy in obtaining impact pressures and forces are the main difficulties of these numerical techniques. Beside mesh-based numerical methods, there are also relevant studies on the modeling of oscillatory free-surface flows with mesh-free methods such as Smoothed Particles Hydrodynamics (SPH) [14, 15] and Moving Particle Semi-implicit (MPS) methods [16]. Khayyer and Gotoh performed multi-phase Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use Journal of Fluids Engineering. Received November 23, 2016; Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242 Copyright (c) 2017 by ASME modeling of sloshing flows using the MPS method [17]. Different surge and pitch enforced sloshing problems were studied by SPH method in literature [18–22]. However, these studies are generally limited to a few critical motion frequencies which do not fully represent the wave response of the fluid motion. The construction of a wave response curve through performing simulations at various excitation frequencies where higher order harmonic modes of fluid motion become effective is a very important ed ite d benchmark for testing the capabilities of numerical methods in terms of capturing long-term sloshing phenomena under a wide variety of physical conditions. py The long term sloshing problem can exhibit different physical behaviors. For example, the fullness Co ratio of the tank can lead to shallow and deep water configurations which might have distinct sloshing characteristic under a given excitation frequency. Moreover, in engineering applications, the higher ot order modes of sloshing in long-term excitations can be significant since these modes may overlap with tN the lowest resonance frequency [11] which induces a damping effect on the evolution of the fluid flow sc rip and causes a remarkable change in the physical behavior of the fluid motion. Towards this end, in this study, three different test cases are chosen carefully to assess the performance of the proposed numerical scheme. The first case involves the comparison of free surface elevation time series on the side wall Ma nu of the rectangular tank which is excited close to resonance frequency. The obtained results for two different fullness ratios of the tank corresponding to shallow and relatively deep water configurations are compared with the experimental and numerical solutions presented in [23]. Having showed that the ed presented weakly compressible SPH (WCSPH) scheme can handle the steady-state wave regimes for both shallow and deep water sloshing in a rectangular tank around corresponding resonance frequencies, pt for completeness, it would be prudent to study in detail the higher order modes of sloshing phenomenon. ce To this end, the rectangular tank with a constant water depth is enforced to move with a series of Ac excitation frequencies and a quantitative comparison on the wave response curves with the results of [11] is given. This part is generally missing in particle based numerical studies in the literature [18,19,21,22] and this work would like to enclose the validation/verification examination of the presented numerical modeling by such a challenging test. To examine the capability of the presented WCSPH scheme for modeling dynamic characteristics of sloshing flows, as a final test case, pressure loads on some critical points of the tank are calculated and compared with the results of the study in [13]. Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use Journal of Fluids Engineering. Received November 23, 2016; Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242 Copyright (c) 2017 ASME Thisbystudy presents a robust and accurate numerical WCSPH scheme capable of capturing sloshing phenomena under a wide variety of physical conditions and contributes to the state of the art in the SPH field in terms of: (i) suggesting a new velocity variance based free surface (VFS) and artificial particle displacement (APD) algorithms to produce a robust and accurate scheme for numerical simulation of violent free-surface flows, (ii) investigating a variety of oscillatory motion conditions and parameters, ed ite d and (iii) performing wave response analysis. The rest of this paper is structured as follows. In section 2, governing equations, SPH discretization schemes and the new free surface and APD algorithms are succinctly explained. In section 3, the simulation results are presented and quantitatively compared with py the experimental and numerical findings in literature in terms of flow kinematics and dynamics. Finally, 2.1 GOVERNING EQUATIONS AND NUMERICAL MODELING sc rip 2 tN ot Co the concluding remarks are drawn in section 4. Field Equations with WCSPH Discretization Ma nu The sway-sloshing problem, characterized as violent free surface flow, is solved using Euler’s equation of motion and continuity coupled with Lagrangian particle advection. Neglecting viscous effects 1 D~u = − ∇P +~g Dt ρ (1) Dρ = −ρ∇ ·~u Dt (2) Ac ce pt ed and allowing for rotation of fluid elements, the governing equations can be written as ~u = D~r Dt Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use (3) Journal of Fluids Engineering. Received November 23, 2016; Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242 Copyright (c) 2017D/Dt,∇, by ASME~u,~r, P and ρ are the material time derivative, nabla operator, velocity and position vecwhere ed ite d tors, pressure and the density of particles, respectively, while~g denotes gravitational acceleration vector. The governing equations are discretized using WCSPH approach. WCSPH method uses an artificial equation of state which couples pressure and density variations through an artificial velocity commonly known as the speed of sound which is lower than the real one under the weakly-compressibility limit. py There are various forms of artificial equation of state used within the scope of WCSPH approach to be Co able to calculate the pressure for computing pressure gradient term in the equation of motion. The one ρ ρ0 γ tN −1 , (4) sc rip ρ0 c20 p= γ ot proposed in [24], is used in this study where c0 is the reference speed of sound, γ is the specific heat-ratio of water and Ma nu is taken equal to 7 while ρ0 is the reference density which is equal to 1000 (kg/m3 ) for fresh water. The value of reference speed of sound is determined by Mach (M) number (a dimensionless quantity representing the ratio of velocity of fluid to speed of sound). It is required that M should be in the ed vicinity of 0.1 in order to keep density variation within %1 of reference density [24]. As a Lagrangian method, SPH represents the flow field by a finite number of moving particles. These particles carry pt the characteristic properties of the flow such as mass, position, velocity, momentum, and energy. SPH ce is basically an interpolation process where the fluid domain is modeled through the interactions of Ac neighboring particles using an analytical function widely referred to as the kernel/weighting function W (rij , h). Here, h is the smoothing length and rij represents the magnitude of the distance vector given as ~rij =~ri −~rj for particle of interest and its neighboring particle, denoted using boldface subscripts i and j, respectively while~ri and~rj are the position vectors for the particles. Although it is computationally more expensive than some of the other kernel functions, the piecewise quintic kernel function is chosen due to its better accuracy and stability characteristics [25]. The compactly supported, two dimensional Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use Journal of Fluids Engineering. Received November 23, 2016; Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242 Copyright (c) 2017 by ASME piecewise quintic kernel function is given as (5) ed ite d (3 − R)5 − 6 (2 − R)5 + 15 (1 − R)5 , 0 ≤ R < 1 (3 − R)5 − 6 (2 − R)5 , 1≤R<2 W (R, h) = αd (3 − R)5 , 2≤R<3 0, R≥3 where R = rij /h and αd is a coefficient dependent on the dimension of the problem [26]. In two di- py mensions, αd is equal to 7/(478πh2 ). The SPH method interpolates any arbitrary continuous function, Z (6) tN Ω A ~rj W (rij , h)d 3~rij , ot Ai ∼ = hA (~ri )i ≡ Co A(~ri ), or concisely denoted as Ai in the following manner: sc rip where the angled bracket hi denotes the kernel approximation, d 3~rij is the infinitesimally small volume element inside the domain and Ω represents the total bounded volume of the domain. The governing equations are discretized based on the above given SPH approximation where the integral operation Ma nu over the volume of the bounded domain is replaced by the summation operation over all neighboring particles j of the particle of interest i. The differential volume element d 3~rij is also replaced by mj /ρj . As a result, the Euler’s equation of motion and the mass conservation may be discretized by the SPH N pi pj d~ui = − ∑ ( 2 + 2 + Πij )∇iWij , dt ρj j=1 ρi (7) N m dρi j ~ui −~uj · ∇iWij . = ρi ∑ dt j=1 ρj (8) Ac ce pt ed method to provide the following relations [24]: Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use Journal of Fluids Engineering. Received November 23, 2016; Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242 Copyright (c) 2017mbydenotes ASME the mass of particle j, ∇ is the gradient operator where the particle identifier i indicates Here, j i that the spatial derivative is evaluated at particle position i. ∏ij is the artificial viscosity term defined as i 0, j (9) ~uij ·~rij ≥ 0 ed ite d Πij = −αµij ci +cj , ~uij ·~rij < 0 ρ +ρ (10) ot Co ~ui −~uj · ~ri −~rj µij = h . ~ri −~rj 2 + θh2 py where (γ−1)/2 tN where, the local speed of sound for a particle, included in Eq. 9 is computed according to ci = co (ρi /ρo ) Artificial viscosity term is inserted into the linear momentum balance equation for inviscid flow to im- sc rip prove the stability of the numerical solution via the addition of a rather small amount of diffusion to the fluid. Artificial viscosity term was initially introduced by [27] for finite difference schemes and has Ma nu been extensively utilized in many SPH studies as well [2]. The level of the artificial viscosity added to the fluid should be optimized to minimize its impact on the solution while preserving its stabilizing effect. It should be noted as the spatial resolution increases (i.e. smoothing length parameter (h) goes to zero), the effect of artificial viscosity goes to zero hence recovering the Euler’s equation of motion [28]. ed Here, θ is a constant which is added to the denominator of the Eq. 10 to prevent any singularity and pt α parameter is the coefficient which determines the amount of the artificial diffusion in the numerical ce solution. The numerical value of α in this study is determined through referring to numerical solutions Ac of benchmark problems. Assigning α = 0.006 is found to provide satisfactory results and used in all of the simulations presented in this work. Time marching of the numerical solution is achieved via a predictor-corrector time integration scheme while free-slip wall boundary condition is applied to all bounding walls through ghost particle technique. The dynamic free-surface condition on the free surface is satisfied by setting the pressure of free surface particles to zero or, equivalently, setting the densities of the free surface particles to the reference Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use . Journal of Fluids Engineering. Received November 23, 2016; Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242 Copyright (c) 2017 byFurther ASME details of the utilized numerical scheme and boundary implementations can be found density. in our previous study [29]. 2.2 Corrective Numerical Treatments Being a Lagrangian numerical method, SPH needs suitable corrective numerical treatments to re- ed ite d duce particle clustering and noisy pressure field which leads to random and rapid pressure oscillations thereby causing reduced numerical accuracy and stability or even break down of the numerical simulations. Here, in addition to well-known density correction algorithm, velocity variance based free surface (VFS) and artificial particle displacement (APD) algorithms are incorporated into the numerical scheme py of WCSPH in a hybrid manner (namely, VFS+APD) to circumvent particle clustering induced numeri- Co cal problems. ot In WCSPH approach one of the main difficulties arises from the occurrence of highly noisy pressure tN field due to the acoustic components derived from the use of an artificial speed of sound. In the recent sc rip years, several studies were addressed to smooth out the density field at each time step using a diffusive term in the continuity equation [18, 30]. A good comprehensive review on limits and capabilities of diffusive terms in literature can be found in [31,32]. In our previous work [29], the effect of APD which Ma nu provides a homogeneous particle distribution on the problem domain and hence helps to prevent acoustic components derived from the use of an artificial speed of sound which leads to a noisy and in accurate pressure field was presented. The proposed numerical scheme is further tested for violent sway-sloshing ed flows together with a variety of different fullness ratio of tank and under the effects of variable enforced pt motion frequencies. Due to the extreme computational costs of density diffusion algorithm [33] and on ce the light of the benchmark simulation results, it is decided to utilize the basic Shepard density smoothing algorithm together with the APD treatment which brings a satisfactory accuracy in terms of kinematic Ac and dynamic flow characteristics. The density correction treatment is applied through b ρi = ρi − σ ∑Nj=1 ρi − ρj Wij ∑Nj=1 Wij , (11) where b ρ is the corrected density, N is the number of neighbor particles for particle i and σ is a constant Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use Journal of Fluids Engineering. Received November 23, 2016; Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242 Copyright (c) 2017is byset ASME which to unity in this study thereby leading to the well-known “Shephard” interpolation [34]. N δ~ri = , bi =~ui − εδ~ui ~u py ∑Nj=1 Wij ~rij ∑ r3 ro2uc f f ∆t, (12) (13) ot j=1 ij Co δ~ui = ∑Nj=1 ~ui −~uj Wij ed ite d The hybrid VFS+APD treatment is formulated as tN where ~ubi is called the corrected velocity used to replace the velocity of the free-surface particle calcu- sc rip lated based on the solution of momentum balance equation at each time step, ε is a constant parameter which is sufficient to assign a value between 0.05-0.1 times the initial particle spacing (dx) and a value of 0.003 is chosen for the cases simulated in the present work. ro is an average distance calculated for each Ma nu particle as ro = ∑j rij /N and uc f f is the velocity based APD coefficient. In the hybrid approach, VFS is applied only to free-surface particles while APD is used to regularize the arrangement of particles with fully populated influence domain, which are identified via the kernel truncation. Namely, particles ed with fully populated influence domain are considered to be internal particles while those whose neighbor numbers are below certain threshold compared to the average neighbor number of all particles with pt fully populated compact support are regarded as free-surface particles. We have experienced that parti- ce cles with neighbor numbers fewer than (60-70)% of the average neighbor particle number demarcates Ac the free-surface accurately. If this ratio is taken lower than 60% we observed that the free surface was not demarcated totally and discontinuities occurred along the free surface profile. On the other hand, if this ratio is assigned more than 70%, the free-surface thickness increases and it leads to unphysical free surface tension forces. Figure 1(a) shows a snapshot of test case 1-b (the simulation parameters are given in the following section Tab. 1) and illustrates the regions where the VFS and APD treatments are effective. Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use Journal of Fluids Engineering. Received November 23, 2016; Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242 Copyright (c) 2017 by ASME The APD formulation employed in this study is different from that in literature [29] such that it uses the magnitude of the velocity variance computed for each particle as a velocity based APD coefficient (herein, referred to as the local velocity coefficient, uc f f = δ~ui ) rather than the magnitude of the maximum velocity in the problem domain (referred to as the global velocity coefficient, uc f f = βumax ) and ed ite d hence eliminates the need for the problem dependent calibration coefficient β in the APD formulation. The existing APD formulation in literature has been reported to be quite successful in yielding homogeneous particle distribution for problems in confined domain. Nevertheless, we observed that once used py in violent free-surface flows such as dam-break and sloshing problems, it leads to the increase in the Co total volume of the water reserve since in these problems, the variation in the velocity field of the flow is extremely high, especially upon and after the impact of the water reserve on the tank walls. Hence, ot utilizing global velocity coefficient in APD requires tailoring the amount of allowable artificial particle tN displacement carefully to avoid the gradual increase of the volume due to the movement of particles toward free-surface. To ensure the conservation of total volume, a long-term simulation of dam-break sc rip problem is performed utilizing exactly the same problem parameters reported our previous work [29]. Figure 1(b) indicates that the employed corrective algorithm (VFS and the new APD algorithm) con- NUMERICAL RESULTS ed 3 Ma nu serves the total volume during the simulation. In this work, sway-sloshing problem is simulated for three different test cases where the first and pt the second test cases address the kinematic characteristics of the sway-sloshing motion while the third ce case investigates the dynamic properties of the fluid motion. The harmonic sway motion of the tank is Ac induced by a sinusoidal function such that x(t) = A sin(ωt) where x is the horizontal position of the tank, A is the amplitude and ω is the angular frequency of the enforced motion. The representative geometry of the problem is given in Fig. 2, and dimensions, amplitudes and angular frequencies of all test cases are tabulated in Tab. 1 where frequency values in test case 2 indicates the minimum and maximum of the investigated frequency range. The theoretical nth natural frequency at a given water depth (D) and tank width (L) is given as [13, 23] Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use Journal of Fluids Engineering. Received November 23, 2016; Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242 Copyright (c) 2017 by ASME r (14) ed ite d ωn = nπ ngπ tanh D . L L For all test cases, the speed of sound, reference density, gravitational acceleration and the smoothing length parameter are taken as c = 40 (m/s), ρ = 1000 (kg/m3 ), g = 9.81 (m/s2 ) and h = 1.33∆x (m), respectively. For the sake of comparison, the initial conditions are taken the same as the reference Co py studies [13, 23] where all particles are at rest and have only hydrostatic pressure at t = 0 (s). Before starting systematical investigations on the kinematic and dynamic characteristics of oscil- ot latory flows in rectangular tanks, a convergence test is performed in order to determine correct par- tN ticle resolution bearing in mind the trade off between the numerical accuracy and the computational sc rip time. Test case 1-a is chosen for the convergence study, and three different particle resolutions, namely, H/dx=15, H/dx=30 and H/dx=60 are simulated. The free-surface elevation, ξ(t), and corresponding frequency domain analysis for each particle resolution is compared with the experimental data of [23] Ma nu which is given in Fig. 3. It can be seen from Fig. 3 that the free-surface elevations at the left wall are in alignment with experimental measurements, especially when the resolution is higher than H/dx=15. The standard deviations of each particle resolutions with respect to experimental measurements are cal- ed culated as 0.072, 0.061 and 0.054 for the spatial resolutions H/dx = 15, 30, 60, respectively. On the other hand, the normalized computational time costs for our serial in-house code on an Intel(R) Xeon(R) CPU pt E5-2690 v3 @2.60GHz processor are approximately, 0.13, 1.0 and 13.3 accordingly. Considering the ce computational cost and the sufficient accuracy of the results, we decided to use the particle resolution Ac of H/dx=30, which approximatively corresponds to the initial particle spacing values ∆x = ∆y = 0.01 (m), and the time step value is chosen as ∆t = 0.00015 (s) for the free-surface elevation simulations performed in test cases 1 and 2. Unlike the examination of free-surface elevations on the tank walls, the calculation of impact pressures on the certain points of tank requires more higher resolution on the problem domain. To capture the pressure variations on the tank walls accurately in test case 3, both spatial and temporal resolutions of the simulation are doubled with respect to cases 1 and 2. Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use Journal of Fluids Engineering. Received November 23, 2016; Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242 Copyright (c) by ASME 3.12017Examination of the Free-Surface Elevations on the Side Wall Investigating the variation of the free-surface profiles during the motion sheds light on the physics behind the oscillatory motion of the fluid in the tank. The first case models sway sloshing with two different tank fullness ratio and is dedicated to the study of kinematic characteristics of the oscillatory ed ite d motion. The Case 1-a is regarded as a shallow water condition where the water depth to tank width ratio D/L=0.173 is lower than the value of 0.24 [12]. For both Case 1-a and 1-b, the angular frequency of the enforced harmonic sway motion is chosen close to the first theoretical natural frequency of the sloshing motion (ω1 in Tab. 1) following the study of [23]. As the tank is enforced to move in an harmonic py manner with a constant frequency, the fluid content is also expected to respond with approximately the Co same frequency for the free-surface elevation on the side walls. The results of the present simulations are compared with the findings of [23] obtained using conventional WCSPH method which employs ot periodical density re-initialization procedure to increase the accuracy of the pressure field, particle re- tN finement algorithm at the tank corners, XSPH numerical treatment and Runge-Kutta time integration sc rip scheme for the time marching of the flow. Fig. 4(a) and Fig. 5(a) present time histories of the water level elevation comparison for two dif- Ma nu ferent cases, namely, Case 1-a and Case 1-b at the given water depth, amplitude and frequency of the enforced motion. It may be noticed that the periods of the fluid motion are in good agreement with both experimental and numerical results of reference study. These results of both test cases are also compared ed with literature in frequency domain, which are given in Fig. 4(b) and Fig. 5(b). Except for a few instances, the amplitudes of the harmonic motion match with the experimental data. These discrepancies pt may be attributed to inherent differences in the determination of free-surface location (on vertical walls ce via particle positions) compared to the data acquired by the probe used in experiments. An additional Ac level of complexity arises when the water splashes after its impact on the walls. In the present work, the numerical data sampling-rate frequency is 11.11 (Hz) and the free-surface is obtained as the average elevation of the free-surface fluid particles within x = −0.805 (m) to x = −0.825 (m), an interval chosen to encompass the physical region covered by the wave probe (x = −0.815 (m)) in the reference study. It should be noted that the specified horizontal positions in the determination of the free surface are given for the initial position of the tank (t = 0(s)). Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use Journal of Fluids Engineering. Received November 23, 2016; Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242 Copyright (c) by ASME 3.22017Examination of Wave Responses Due to Enforced Motion Frequencies The response of higher order modes of sloshing phenomenon should also be investigated where these modes can be significantly important when they overlap with the lowest resonance frequency [11]. Higher order modes and especially the second harmonics of the motion can be effective in long-term excitations and may change the physical behavior of the fluid motion. To test the capabilities of the ed ite d proposed numerical scheme further, the rectangular tank with constant water depth is enforced to move with a series of excitation frequencies. Excitation frequencies are 6.26, 5.63, 5.21, 4.81, 4.54, 4.32, 4.19, 4.08, 3.98, 3.78, 3.67, 3.62, 3.50 py (rad/s). The simulation results are presented and compared with literature [11] in the form of wave response curve as shown in Fig. 6 where T and T1 denote the excitation period and the first theoretical Co natural period of the motion, respectively. Our results are in better agreement with the experimental ot measurements of reference work in the range between 0.8 < T /T1 < 1.1 than the simulation results pre- tN sented within the same study. In the frequency interval of 1.1 < T /T1 < 1.25, the secondary resonance mode becomes effective leading to a notable drop in wave amplitude which is captured with a slight sc rip delay in our simulation. As the excitation frequency increases, the higher order modes become prevalent thereby further decreasing the wave elevation which is computed within the same order of accuracy 3.3 Ma nu with the numerical results of the reference study in comparison to the experimental data [11]. Examination of the Impact Pressures on the Side and Top Surfaces of the Tank ed Having obtained simulation results in mesh with those reported in [23] and [11] for kinematic characteristics of the sway-sloshing motion, the ability of the proposed scheme is tested if it can capture pt the more sensitive dynamic of the flow. The results are compared with level set simulations of [13]. ce The impact pressure time series of the water reserve are calculated at four different points of the tank, Ac designated as P1 through P4 in Fig. 7. As the impact pressure on the walls may change in small fractions of a second due to passing waves and splashes, simulation results are saved every 0.0075 (s) which corresponds to a data capture frequency of 133.33 (Hz). Pressure values at the given sensor points of the tank are interpolated by using the same quintic kernel function utilized during the simulations. However, the smoothing length is set Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use Journal of Fluids Engineering. Received November 23, 2016; Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242 Copyright (c) by ASME to 2017 be four times bigger than the value used in simulations to increase the contributing particle number, hence accounting for the pressure sensor’s diameter and filter out local oscillations in pressure. Ghost particles generated during the implementation of the wall boundary conditions are also included in the interpolation processes at all given pressure sensor points on the tank surface. ed ite d The comparison of the full time history of pressures with the experimental and numerical results of the reference data is shown in Figs. 8 and 10 for pressure points P1 and P4 , respectively. Table 2 provides the consecutive maximum pressure values due to the harmonic motion of the fluid for pressure points py P2 and P3 . Co The first pressure point P1 remains below free-surface level during the course of flow evolution, yielding a continuous and more reliable pressure data acquisition point from both numerical and exper- ot imental points of view. On the other hand, measuring inherently noisy pressure values in experimental tN studies generally requires a tedious procedure of post-processes like signal processing or data filtering in order to obtain reasonable analyses. This situation is also reported in the referenced study [13] such sc rip that the authors processed the given face pressure values because the experimental set-up does not give any information whether there is a signal processing nor data filtering over the input data. Keeping this Ma nu uncertainty on the experimental results in mind, continuous pressure time history of the present simulation provided in Fig. 8, is able to capture the wiggly nature of the pressure data collected by the P1 sensor. The obtained results are also in match with period and magnitude of the pressure peaks of the pt engineering applications. ed experimental data which constitute critical loads for structural optimization and the point of interest for ce The pressure points P2 , P3 and P4 do not sense the fluid pressure continuously as P2 lies just above Ac the initial free-surface level and P3 and P4 are close to the top right corner of the tank as shown in Fig. 7. Unlike the continuously varying pressure profile obtained for P1 location, P2 , P3 and P4 locations will register pressure data as periodical pressure beats which last over very small time intervals. The comparative results for the data points P2 and P3 are given as consecutive maximum pressure values in Tab. 2. It is observed from Tab. 2 that the standard deviation of the experimental measurements of the ref- Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use Journal of Fluids Engineering. Received November 23, 2016; Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242 Copyright (c) 2017 by ASME erence data is higher than the numerical results, which is expected due to the difficulty in collecting precise data from the pressure sensors. If mean value and standard deviation of these consecutive impact pressures are taken into account together, one may practically infer that the SPH impact pressures are compatible with those of the experimental measurements. The standard deviations of both numerical results with respect to the experimental measurements are also performed and found as follows: For the ed ite d hit point P2 , SPH results have a standard deviation of 1.55 while [13] has 2.71. For the hit point P3 , SPH results have a standard deviation of 4.89 while referenced study has 6.54. To show the consecutive pressure hits at sensor locations P2 and P3 clearly, Fig. 9 is also provided where the horizontal lines py denotes the mean values of the pressure beats. The consecutive pressure beats of the present SPH study Co is between the numerical and experimental results of [13]. ot To illustrate that the current SPH algorithm is also able to capture the pressure beats in extreme tN sensor locations on the tank, the pressure time history at P4 sensor location is displayed together with the corresponding experimental and numerical results of the associated literature in Fig. 10. As there sc rip are sudden and rapid impacts which occur less than 0.025s in real time, it requires very sensitive and accurate data processing on sensor points P3 and P4 and lead to high memory necessities and rise in Ma nu computational time costs. Despite of utilizing a high data-sampling value of 133.33Hz, it should be noted that there may have some missing instants during the small impact time interval while reading pressure time series. Nevertheless, if the periodicity and amplitude of pressure beats of water reserve are taken into account, the obtained results of the present study are compatible with those reported in pt ed numerical and experimental results of reference study. ce A closer survey on the evolution of a single pressure beat on the left wall is also performed in Ac order to observe the complex dynamics of the fluid motion. The instantaneous pressure beats on the wall may have a prominent importance in the strength of the solid tank because the superposition of these pressure beats determines the magnitude of the critical loads on the structure. Towards this end, a period of single pressure beat which is calculated at P1 is divided into four separate instants, namely, t1 = 10.54(s),t2 = 10.64(s),t3 = 10.86(s) and t4 = 11.17(s) which were shown in Fig. 8(b). To explain the fluid flow between these time fractions in a more clear way, the free-surface profiles and the pressure Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use Journal of Fluids Engineering. Received November 23, 2016; Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242 Copyright (c) 2017ofbythe ASME fields whole domain at the given instants are also shown in Fig. 11. It can be said from the Fig. 8 that the proposed SPH method is able to capture the primary and secondary pressure peaks on a given main pressure peak as observed in the experimental data. The main peak in the pressure occurs when a period of the harmonic motion is about to finish (t1 ) and the velocity of the tank is converging to zero while at the same time the fluid particles are attacking to the vertical wall with a very high speed ed ite d which yields a violent impact on the wall. As soon as the tank changes its direction a sudden drop in pressure occurs (t2 ) which is then followed by a secondary pressure peak (t3 ) when the velocity of the rigid tank reaches its maximum velocity in the opposite direction. Eventually, the water reserve moves py to the right wall with an increasing velocity, the pressure starts to reduce on the left wall and finally gets CONCLUSION ot 4 Co its minimum value at t4 when the water level at the left wall is minimum. tN In this paper, a modeling study for two-dimensional oscillatory motion of partially filled rectangular tanks is presented where the test cases are chosen to target different physics in sloshing; namely, sc rip (i) the resonance conditions of two different fullness ratios, (ii) wave response analysis based on the series of excitation motion frequency, and (iii) dynamic analysis of fluid behavior by probing pressure Ma nu field at extreme locations of tank. Euler’s equation of motion and continuity equation are discretized using WCSPH approach and integrated in time via a predictor-corrector scheme. Density correction and hybrid VFS+APD numerical treatments are incorporated into the conventional WCSPH numerical solu- ed tion procedures to improve the accuracy and robustness of the numerical solution. Long-term dam-break simulations are carried out using a new APD algorithm with the local velocity coefficient computed for pt each particle and it is shown that the total volume of the fluid domain is exactly conserved even in the Ac ce long-term simulation of dam-break problem. Having showed the effectiveness of the local velocity coefficient in the APD algorithm in terms of giving uniform particle distribution while conserving total volume, the kinematic and dynamic flow characteristics of sway-sloshing motion are investigated for three different cases. The first case has two benchmark problems which are modeled for the same tank geometry with different water depths and motion frequencies. The free-surface elevations on the left wall of the tank are compared with Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use Journal of Fluids Engineering. Received November 23, 2016; Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242 Copyright (c) 2017 by ASME experimental and numerical solutions available in literature, showing good agreement in both time and frequency domains. The second case tests the capability of the presented SPH scheme for a wide range of enforced motion frequencies which yields the response of fluid motion in different harmonic modes. In light of results of the wave response analysis, it is shown that the proposed SPH algorithm captures the general characteristics of the oscillatory motion and shows a good agreement with the experimental and ed ite d computational results found in the literature. The last case analyzes and compares the impact pressures on four points of the tank available in literature data. Comparison of fully submerged sensor point’s pressure values shows satisfactory match with experimental data. As for the points that have occasional py contact with fluid, the mean and standard deviations of pressure fields of this study are compatible with Co experimental data. ot Considering the results obtained fr all three test cases, it is possible to infer that the utilized WCSPH tN scheme and related corrective numerical treatments are able to capture the physics behind the evolution of oscillatory motion in partially filled rectangular tanks. On the enlightenment of the simulation results, sc rip it is also possible to input obtained hydrodynamics loads acting on walls for the structural design and optimization studies in future, which can be useful for practical engineering applications like ships 5 ACKNOWLEDGMENT Ma nu having large liquid cargo tanks and/or liquid containers on board. The authors gratefully acknowledge financial support provided by the Scientific and Technological Ac ce pt ed Research Council of Turkey (TUBITAK) for project number 112M721. Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use Journal of Fluids Engineering. Received November 23, 2016; Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242 Copyright (c) 2017 by ASME References [1] Lee, S., Kim, M., Lee, D., Kim, J., and Kim, Y., 2007. “The Effect of LNG Tank Sloshing on the Global Motions of LNG Carriers”. Ocean Engineering, 34, pp. 10–20. [2] Delorme, L., Iglesias, S., and Perez, S., 2005. “Sloshing loads simulation in lng tankers with sph”. In Proceedings of International Conference on Computational Methods in Marine Engineering, ed ite d MARINE, pp. 1–10. [3] Delorme, L., Colagrossi, A., Iglesias, S., Rodriguez, R., and Botia-Vera, E., 2009. “A Set of Canonical Problems in Sloshing, Part I: Pressure Field in Forced Roll Comparison Between Experimental py Results and SPH”. Ocean Engineering, 36, pp. 168–178. [4] Servan-Camas, B., Cercos-Pita, J. L., Garcia-Espinoza, J., and Souto-Iglesias, A., 2016. “Time Co Domain Simulation of Coupled Sloshing Seakeeping Problems by SPH FEM Coupling”. Ocean ot Engineering, 123(7), pp. 383–396. tN [5] Faltinsen, O. M., and Timokha, A. N., 2009. Sloshing. Cambridge University Press. [6] Okamoto, T., and Kawahara, M., 1990. “Two Dimensional Sloshing Analysis by Lagrangian Finite sc rip Element Method”. International Journal of Numerical Methods in Fluids, 11, pp. 453–477. [7] Wang, C., Teng, J., and Huang, G., 2011. “Numerical Simulation of Sloshing Motion Inside a Two Dimensional Rectangular Tank by Level Set Method”. International Journal of Numerical Ma nu Methods for Heat & Fluid Flow, 21(1), pp. 5–31. [8] Guorong, Y., Subhask, R., and Kamran, S., 2009. “Experimental Study of Liquid Slosh Dynamics in a Partially-Filled Tank”. ASME J. Fluids Eng., 131(7), pp. Article ID 071303, 14 pages. ed [9] Shin, H. R., 2005. “Unstructured Grid Based Reynolds-Averaged Navier-Stokes Method for Liquid pt Tank Sloshing”. ASME J. Fluids Eng., 127, pp. 572–582. ce [10] Gotoh, H., Khayyer, A., Ikari, H., Arikawa, T., and Shimosako, K., 2014. “On Enhancement of Ac Incompressible SPH Method for Simulation of Violent Sloshing Flows”. Applied Ocean Research, 46, pp. 104–115. [11] Faltinsen, O., and Timokha, A., 2001. “An Adaptive Multimodal Approach to Nonlinear Sloshing in a Rectangular Tank”. Journal of Fluid Mechanics, 432, pp. 167–200. [12] Faltinsen, O., and Timokha, A., 2002. “Asymptotic Modal Approximation of Nonlinear Resonant Sloshing in a Rectangular Tank with Small Fluid Depth”. Journal of Fluid Mechanics, 470, Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use Journal of Fluids Engineering. Received November 23, 2016; Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242 Copyright (c) 2017 ASME pp.by319–357. [13] Chen, Y., Djidjeli, K., and Price, W., 2009. “Numerical Simulation of Liquid Sloshing Phenomena in Partially Filled Containers”. Computers & Fluids, 38, pp. 830–842. [14] Monaghan, J., and Gingold, R., 1977. “Smoothed Particle Hydrodynamics: Theory and Application to Non-Spherical Stars”. Monthly Notices of The Royal Astronomical Society, 181, pp. 375– ed ite d 389. [15] Lucy, L., 1977. “Numerical Approach to Testing the Fission Hypothesis”. Astronomical Journal, 82, pp. 1013–1024. py [16] Koshizuka, S., and Oka, Y., 1996. “Moving Particle Semi-implicit Method for Fragmentation of Incompressible Fluid, nuclear science and engineering”. Nuclear Science and Engineering, 123, Co pp. 421–434. ot [17] Khayyer, A., and Gotoh, H., 2013. “Enhancement of Performance and Stability of MPS Mesh- Computational Physics, 242, pp. 211–233. tN free Particle Method for Multiphase Flows Characterized by High Density Ratios”. Journal of sc rip [18] Antuono, M., Colagrossi, A., Marrone, S., and Molteni, D., 2010. “Free Surface Flows Solved by Means of SPH Schemes with Numerical Diffusive Term”. Computer Physics Communications, 181, pp. 532–549. Ma nu [19] Rafiee, A., Cummins, S., Rudman, M., and Thiagarajan, K., 2012. “Comparative Study on the Accuracy and Stability of SPH Schemes in Simulating Energetic Free Surface Flows”. European Journal of Mechanics B/Fluids, 36, pp. 1–16. ed [20] Chowdhurry, S., and Sannasiraj, S., 2014. “Numerical Simulation of 2D Sloshing Waves Using pt SPH with Diffusive Terms”. Applied Ocean Research, 47, pp. 219–240. ce [21] Cao, X., Ming, F., and Zhang, A., 2011. “Sloshing in a Rectangular Tank Based on SPH Simula- Ac tion”. Computer Physics Communications, 182, pp. 866–877. [22] Meringolo, D., Colagrossi, A., Marrone, S., and Aristodemo, F., 2017. “On the Filtering of Acoustic Components in Weakly-Compressible SPH Simulations”. Journal of Fluids and Structures, 70, pp. 1–23. [23] Pakozdi, C., 2008. “A Smoothed Particle Hydrodynamics Study of Two-Dimensional Nonlinear Sloshing in Rectangular Tanks”. PhD thesis, Norwegian University of Science and Technology, Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use Journal of Fluids Engineering. Received November 23, 2016; Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242 Copyright (c) 2017 by ASME University of Science and Technology. Norwegian [24] Monaghan, J., and Kos, A., 1999. “Solitary Waves on a Cretan Beach”. Journal of Waterway Port Coastal and Ocean Engineering-Asce, 125(3), pp. 145–154. [25] Shadloo, M., Zainali, A., Sadek, H., and Yildiz, M., 2011. “Improved Incompressible Smoothed Particle Hydrodynamics Method for Simulating Flow Around Bluff Bodies”. Computational Meth- ed ite d ods in Applied Mechanical Engineering, 200, pp. 1008–1020. [26] Liu, M., and Liu, G., 2010. “Smoothed Particle Hydrodynamics: An Overview and Recent Developments”. Arch. Comput. Methods Eng., 17, pp. 25–76. py [27] von Neumann, J., and Richtmyer, R., 1950. “A Method for the Numerical Calculation of Hydrodynamic Shocks”. Journal of Applied Physics, 21, pp. 232–247. Co [28] Antuono, M., Colagrossi, A., Marrone, S., and Lugni, C., 2011. “Propagation of Gravity Waves ot Through an SPH Scheme with Numerical Diffusive Terms”. Computer Physics Communications, tN 182, pp. 866–877. [29] Ozbulut, M., Yildiz, M., and Goren, O., 2014. “A Numerical Investigation into the Correction Mechanical Sciences, 79, pp. 56–65. sc rip Algorithms for SPH Method in Modeling Violent Free Surface Flows”. International Journal of [30] Antuono, M., Colagrossi, A., and Marrone, S., 2012. “Numerical Diffusive Terms in Weakly- Ma nu Compressible SPH Schemes”. Computer Physics Communications, 183, pp. 2570–2580. [31] Meringolo, D., Colagrossi, A., Aristodemo, F., and Veltri, P., 2015. “SPH Numerical Modeling of WavePerforated Breakwater Interaction”. Coastal Engineering, 101, pp. 48–68. ed [32] Meringolo, D., Colagrossi, A., Aristodemo, F., and Veltri, P., 2015. “Assessment of Dynamic pt Pressures at Vertical and Perforated Breakwaters Through Diffusive SPH Schemes”. Mathematical ce Problems in Engineering, 2015, p. 10 pages. Article ID 305028. Ac [33] Nomeritae, N., Daly, E., Grimaldi, S., and Bui, H., 2016. “Explicit Incompressible SPH Algorithm for Free-Surface Flow Modelling: A Comparison with Weakly Compressible Schemes”. Advances in Water Resources, 97, pp. 156–167. [34] Shepard, D., 1968. “A two dimensional interpolation function for irregularly-spaced data”. In Proceedings of the 23rdACM National conference, pp. 517–524. Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use Ac ce pt ed Ma nu sc rip tN ot Co py ed ite d Journal of Fluids Engineering. Received November 23, 2016; Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242 Copyright (c) by ASME 6 2017 TABLES Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use Co py ed ite d Journal of Fluids Engineering. Received November 23, 2016; Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242 Copyright (c) 2017 by ASME H (m) D (m) L (m) ot Dimension, amplitude and frequency values for the simulated test cases A (m) ω (rad/s) ω/ω1 0.08 3.488 1.173 0.08 3.696 1.032 tN Table 1. 1.05 0.30 1.73 Case 1-b 1.05 0.50 1.73 Case 2 1.00 0.35 1.00 0.05 3.50-6.26 0.70-1.26 Case 3 0.50 0.35 0.80 0.02 5.820 1.000 Ac ce pt ed Ma nu sc rip Case 1-a Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use Journal of Fluids Engineering. Received November 23, 2016; Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242 Copyright (c) 2017 by ASME Comparison of consecutive impact pressure values at points P2 and P3 with the experimental and numerical results of [13] Numerical Results (kPa) P3 P2 P3 1st 0.77 9.80 0.46 4.72 2nd 4.30 7.01 0.54 8.01 3rd 8.26 9.11 1.68 6.22 4th 3.28 8.12 2.64 2.93 5th 4.62 10.42 2.78 1.65 6th 2.82 10.83 1.98 2.72 7th 3.91 13.33 1.27 8th 2.50 17.84 1.37 9th 3.13 5.24 1.73 10th 5.11 13.35 1.86 11st 2.75 13.45 12nd 3.88 3.36 13rd 4.30 14th P2 P3 ed ite d P2 0.67 4.49 1.11 5.03 4.14 5.78 2.63 4.54 3.57 5.19 2.37 6.92 3.70 6.32 4.31 5.15 5.35 3.92 2.59 5.41 3.59 2.72 8.27 3.29 2.66 6.11 1.78 3.63 3.35 3.68 5.36 1.92 3.55 3.57 5.89 3.66 8.17 1.97 3.85 2.88 5.73 15th 4.83 7.08 2.07 3.85 3.29 4.70 16th 4.09 9.78 2.03 3.64 3.00 9.14 17th 3.91 4.79 1.97 - 4.55 6.97 18th 4.23 5.09 1.80 - 4.64 5.35 19th 3.66 8.70 2.00 - 3.10 - 3.42 - - - 3.19 - 3.70 - - - 3.34 - 3.86 8.99 1.78 4.00 3.15 5.83 Std. Deviation 1.38 3.68 0.57 1.44 1.05 1.35 Max. Value 8.26 17.84 2.78 8.01 5.15 9.14 ce 21st Ac Average 1.93 sc rip Ma nu ed 20th ot 4.05 pt Co Consecutive Impacts SPH Results (kPa) py Experimental Results (kPa) tN Table 2. Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use Journal of Fluids Engineering. Received November 23, 2016; Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242 Copyright (c) by ASME 7 2017 FIGURES 1.2 1 APD Particles VFS Particles Boundary Particles z (m) 0.8 0.6 0.4 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 x (m) (a) 1 Fluid Particles Boundary Particles Final Volume Level z(m) 0.6 Co 0.4 0.2 0 0.5 1 1.5 x(m) 2.5 3 tN (b) 2 ot 0 Fig. 1. 1.8 py 0.8 ed ite d 0.2 The view of free surface (red) and fully populated (blue) regions during the evolution of sway-sloshing motion (at t ∼ 20(s) (a), and Ac ce pt ed Ma nu sc rip the conservation of final volume at t = 25(s) for the dam-break problem with the local velocity coefficient in the APD treatment (b) Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use The representative geometry for the sway-sloshing simulations Ac ce pt ed Ma nu sc rip tN ot Co py Fig. 2. ed ite d Journal of Fluids Engineering. Received November 23, 2016; Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242 Copyright (c) 2017 by ASME Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use Journal of Fluids Engineering. Received November 23, 2016; Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242 Copyright (c) 2017 by ASME 0.35 Experiment H/dx=15 H/dx=30 H/dx=60 0.3 0.25 (ζ(t)−D)/L 0.2 0.15 0.1 0.05 −0.1 15 20 25 30 35 40 45 50 55 60 t(g/L)0.5 (a) 180 70 py 140 Co 120 |FFT(ζ(t)/L)| 65 Experiment H/δx=15 H/δx=30 H/δx=60 160 100 80 ot 60 20 2 3 4 5 6 tN 40 0 1 ed ite d 0 −0.05 7 8 9 10 11 12 sc rip ω*(L/g)0.5 (b) Fig. 3. Comparison of the three different particle resolutions with experimental results (a) time histories of the free-surface elevations and = −0.815 (m) for Case 1-a Ac ce pt ed Ma nu (b) corresponding frequency analysis at x Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use Journal of Fluids Engineering. Received November 23, 2016; Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242 Copyright (c) 2017 by ASME Pakozdi Experiment Pakozdi SPH Results Present SPH Study 0.4 0.35 0.3 0.25 (ζ(t)−D)/L 0.2 0.15 0.1 0.05 0 −0.1 20 30 40 50 60 70 80 t(g/L)0.5 (a) 150 ed ite d −0.05 90 Co 100 ot |FFT(ζ(t)/L)| py Present SPH Study Pakozdi Experiment Pakozdi SPH Results 1 2 3 4 5 6 7 ω(L/g)0.5 sc rip 0 tN 50 8 9 10 11 12 (b) Fig. 4. Comparison of the (a) time histories of the free-surface elevations and (b) corresponding frequency analysis at Ac ce pt ed Ma nu Case 1-a x = -0.815 (m) for Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use Journal of Fluids Engineering. Received November 23, 2016; Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242 Copyright (c) 2017 by ASME 0.5 Pakozdi Experiment Pakozdi SPH Solution Present SPH Study 0.4 (ζ(t)−D)/L 0.3 0.2 0.1 ed ite d 0 −0.1 −0.2 0 10 20 30 40 t(g/L)0.5 50 60 Present Study SPH Solution Pakozdi Experiment Pakozdi SPH Solution Co 80 70 ot 50 tN |FFT(ζ(t)/L)| 60 40 20 10 2 3 4 5 Ma nu Fig. 5. 1 sc rip 30 0 80 py (a) 70 6 ω (L/g)0.5 7 8 9 10 11 12 (b) Comparison of the (a) time histories of the free-surface elevations and (b) corresponding frequency analysis at the left wall at Ac ce pt ed -0.815 (m) for Case 1-b Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use x= Journal of Fluids Engineering. Received November 23, 2016; Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242 Copyright (c) 2017 by ASME 1 Model Test (Faltinsen & Timokha, 2001) Computational Results (Faltinsen & Timokha, 2001) Present SPH Study 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.7 1 1.1 T/T1 1.2 1.3 1.4 1.5 The comparison of wave elevations near wall with D/L=0.35, A/L=0.05 Ac ce pt ed Ma nu sc rip tN ot Co Fig. 6. 0.9 py 0.8 ed ite d Wave elevation near the wall (m) 0.9 Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use Positions where pressure time series are calculated; dimensions are given in (mm) Ac ce pt ed Ma nu sc rip tN ot Co py Fig. 7. ed ite d Journal of Fluids Engineering. Received November 23, 2016; Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242 Copyright (c) 2017 by ASME Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use Journal of Fluids Engineering. Received November 23, 2016; Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242 Copyright (c) 2017 by ASME 5 4.5 Pressure (kPa) 4 3.5 3 2.5 2 1.5 2 4 6 8 10 12 Time (s) 14 16 (a) Experimental results reported in [13] 5 t1 t 3 Co 3.5 3 t 2 2.5 t4 2 4 6 8 tN 2 1.5 0 20 py 4 18 ot Pressure (kPa) 4.5 ed ite d 1 0.5 0 10 Time (s) 12 14 16 18 20 sc rip (b) Results of the present study 4.5 Ma nu Pressure (kPa) 4 3.5 3 2.5 2 Fig. 8. 4 6 8 10 12 Time(s) 14 16 18 20 (c) Numerical results reported in [13] Comparison of the pressure time histories at P1 Ac ce pt 1.5 0 ed 2 Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use Journal of Fluids Engineering. Received November 23, 2016; Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242 Copyright (c) 2017 by ASME 9 Model Test (Chen et.al. 2009) Computational Results (Chen et.al. 2009) Present SPH Study 8 7 Pressure (kPa) 6 5 4 3 1 0 0 2 4 6 8 10 12 14 16 Number of Consecutive Hits of Water Reserve 18 (a) Consecutive pressure beat at point P2 22 py 18 Co 16 14 12 10 ot Pressure (kPa) 20 Model Test (Chen et.al. 2009) Computational Results (Chen et.al. 2009) Present SPH Study 20 8 tN 6 4 2 4 sc rip 2 0 0 ed ite d 2 6 8 10 12 14 Number of Consecutive Hits of Water Reserve 16 18 (b) Consecutive pressure beat at point P3 Fig. 9. Quantitative comparison of the consecutive hits at pressure points P2 and P3 . Straight lines with the same color indicate the mean Ac ce pt ed Ma nu impact pressure values of the corresponding result Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use Journal of Fluids Engineering. Received November 23, 2016; Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242 Copyright (c) 2017 by ASME 20 Pressure (kPa) 15 10 5 0 2 4 6 8 10 12 Time (s) 14 16 18 (a) Experimental results reported in [13] 8 py Pressure (kPa) 6 Co 4 2 4 6 8 10 Time (s) 12 14 16 18 20 tN 2 ot 0 −2 0 20 ed ite d −5 0 (b) Results of the present study sc rip 12 10 Pressure (kPa) 8 4 2 0 −2 0 Ma nu 6 2 4 6 8 10 12 Time (s) 14 16 18 20 ed (c) Numerical results reported in [13] Comparison of the pressure time histories at P4 Ac ce pt Fig. 10. Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use Journal of Fluids Engineering. Received November 23, 2016; Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242 Copyright (c) 2017 by ASME 3.5 8 0.5 0.5 3 7 2 5 0.3 4 3 0.2 2.5 0.4 6 z (m) z (m) 0.4 0.3 1.5 1 0.2 0.5 2 0.1 0.1 1 0 0 0 −0.5 0 −1 0 0.1 0.2 0.3 0.4 x (m) 0.5 0.6 0.7 −1 0.8 0 0.1 0.2 0.3 0.4 x (m) 0.5 0.6 0.7 0.8 0.5 3.5 0.5 ed ite d (a) Free-surface profile and pressure field at t1 = 10.54(s) (b) Free-surface profile and pressure field at t2 = 10.64(s) 3 0.4 2.5 2 1.5 0.2 1 0.5 0.1 0.3 0.2 0.1 0 0 −0.5 0.1 0.2 0.3 0.4 x (m) 0.5 0.6 0.7 0 0.8 (c) Free-surface profile and pressure field at t3 = 10.86(s) Fig. 11. 0.1 0.2 0.3 0.4 x (m) 0.5 3 2.5 2 1.5 1 0.5 0 −0.5 −1 0.6 0.7 0.8 Co 0 0 py 0.3 z (m) z (m) 0.4 3.5 (d) Free-surface profile and pressure field at t4 = 11.17(s) The snapshots of the free-surface profiles and pressure field between the time interval 10.54-11.17(s). The pressure values in color Ac ce pt ed Ma nu sc rip tN ot bar are given in kPa. 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