Investigation of the Effects of Baffles on the Shallow Water Sloshing in A Rectangular Tank Using A 2D Turbulent ISPH Method

Liquid sloshing is a common phenomenon in the liquid tanks transportation. Liquid waves lead to fluctuating forces on the tank wall. Uncontrolled fluctuations lead to large forces and momentums. Baffles can control these fluctuations. A numerical method, which has been widely used to model this phenomenon, is Smoothed Particle Hydrodynamics (SPH). The Lagrangian nature of this method makes it suitable for simulating free surface flows. In the present study, an accurate Incompressible Smoothed Particle Hydrodynamics (ISPH) method is developed and improved using the kernel gradient correction tensors, particle shifting algorithms, k–ε turbulence model, and free surface particle detectors. Comparisons with the experimental data approve the ability of the present algorithm for simulating shallow water sloshing. The main aim of this study is to investigate the effects of the vertical baffle on the damping of liquid sloshing. Results show that baffles number has a major role in sloshing fluctuation damping.


Introduction
Motion of liquid in containers which has been called sloshing, in fact, is the motion of structure that is transmitted to the liquid. Therefore, it has been categorized as flow structure interaction (FSI). Currently, the interaction of flow and structure is one of the most important problems in some industries. Pumps, turbines, airplanes, and ships are examples of these problems. In this study, this phenomenon has been investigated, experimentally and numerically. Eulerian and Lagrangian numerical methods have been used to simulate FSI problems. Eulerian methods are usually gridbased; therefore, the motion of a solid body grid should be defined and imposed during any iteration. Lagrangian and meshfree methods, such as Smoothed Particle Hydrodynamics (SPH), also could model FSI. SPH was introduced by Lucy (1977), and Gingold and Monaghan (1977). Gingold and Monaghan (1982) used this method to simulate a compressible and inviscid flow, and Morris et al. (1997) used SPH to model incompressible flow.
There are several studies in the multiphase flows Rahmat et al., 2014) and free surface flows (Ozbulut et al., 2017;Shadloo et al., 2015). Newtonian (Shadloo et al., 2011;Sefid et al., 2014), and non-Newtonian flows (Hashemi et al., 2011, Shamsoddini et al., 2015a, free Surface flow (Farrokhpanah et al., 2015), two-phase flow , mixing flow (Shamsoddini et al., , 2015b(Shamsoddini et al., , 2016Shamsoddini and Sefid, 2015) and FSI problems (Hashemi et al., 2011(Hashemi et al., , 2012 are examples in which SPH has widely been used. One of the first works which successfully modeled the fluid-structure interaction (FSI) is the study of Antoci et al. (2007); in their method, both fluid and solid phases were described by smoothing particle hydrodynamics: fluid dynamics was studied in the inviscid approximation, while solid dynamics was simulated through an incremental hypoelastic relation. Han and Hu (2018) modeled the FSI problems in a uniform SPH framework. Zainali et al. (2013) are one of the first scientific teams that used artificial particle shifting (APS) in multi-phase flows. They presented a multiphase incompressible smoothed particle hydrodynamics method with an improved interface treatment procedure for modeling single vortex flow, square droplet deformation, droplet deformation in shear flow, and the Newtonian bubble rising in viscous and viscoelastic liquids. Shadloo et al. (2015) simulated the breaking and non-breaking long waves using SPH. They investigated the effectiveness of SGS turbulent model for violent free surface flows. Ozbulut et al. (2017) simu-lated a two-dimensional oscillatory motion in partially filled rectangular tanks using SPH method which discretizes the governing equations with the velocity variance-based free surface (VFS) and artificial particle displacement (APD) algorithms. They investigated the effects of the tank geometries, fullness ratios, and motion frequencies. More applications of industrial SPH modeling can be found in the study of Shadloo et al. (2016).
In this study, an Incompressible Smoothed Particle Hydrodynamics (ISPH) algorithm is introduced to model a one-way coupled FSI problem with free surface flow. SPH encounters special problems, such as particle clustering, defects, and tensile instability. Therefore, this method evolved from the advanced discretization style, shifting algorithms, turbulence modeling, and free surface particle detecting. This method had also been assessed with the experimental results. Since SPH is a Lagrangian and meshless method, it could model free surface flow easily. Rostami Varnousfaaderani and Ketabdari (2015) presented a modified Weakly Compressible Smoothed Particle Hydrodynamic (WCSPH) method to simulate two-dimensional plunging solitary wave breaking process. Violeau and Issa (2007) examined different turbulence models for SPH simulations. Omidvar and Nikeghbali (2017) applied the SPH method for simulating dam-break propagation over an erodible bed and sediment distribution beneath the steady water flow. These researchers used WCSPH. However, non-physical fluctuations in the density field undermined these studies. WCSPH and ISPH are two well-known SPH methods. ISPH treats the WCSPH method (Lee et al., 2008), but recent efforts on the WCSPH method have resolved the non-physical fluctuations problem (Hashemi et al., 2011;Shamsoddini et al., 2015a). WCSPH uses a state equation to calculate the pressure, while the ISPH method solves the Poisson equation to determine the pressure. ISPH has been selected for the present simulation.
One of the most important problems in free surface flows is liquid sloshing in the tanks, which is a known phenomenon in liquid transport tanks. Sloshing phenomena may create great forces and momentums. Hence, controlling the tank and its carrier may be difficult and unsafe. Therefore, predicting this phenomenon is essential in the liquid transport industries. There are extensive researches in this field. Kim et al. (2012) presented a comparative study on model-scale sloshing tests. Zou et al. (2015) studied the effect of liquid viscosity on sloshing using a set of experiments. Hou et al. (2012) simulated liquid sloshing behavior in a 2D rectangular tank using the ANSYS-FLUENT software. Godderidge et al. (2009) modeled sloshing flow in a rectangular tank with the commercial CFD code. SPH is also a convenient method to model liquid sloshing. There are some researches in this area with SPH method (Cao et al., 2014;Gotoh et al., 2014;De Chowdhury and Sannasiraj, 2014;Shao et al., 2012). ISPH is also an effective method for these cases. Although there are still some valuable cases for study in sloshing phenomena, some aspects of the shallow water sloshing have not been investigated yet. One strategy to reduce the sloshing fluctuations is the baffle mechanism. Therefore, a robust modified explicit ISPH method has been developed and applied for simulating the shallow water sloshing phenomena in a rectangular box. This method is improved using the kernel gradient correction, free surface particle detector, k-ε turbulent model and particle shifting algorithm. Simultaneously, an experimental setup was designed and constructed to record the shallow water sloshing details, experimentally. Comparisons between the numerical and experimental results approve the accuracy of the present algorithm. The main aim of the present study is the investigation of the baffle effects on the controlling of the shallow water sloshing phenomena. Therefore, in the following, first the numerical method and algorithm are introduced, next, the experimental setup is explained and then the results are discussed.

Numerical procedure
SPH has been based on an integral approximation: r r ′ where f(r) is an arbitrary function, h is the smoothing length, W is an interpolation function (kernel function), is the position vector, and is the sub-integral variable. The integral form is approximated using a summation on the discrete points: .
(2) Different interpolation functions were examined and it has been resulted that the fifth-order Wendland kernel function is accurate for modeling fluid flows (Dehnen and Aly, 2012). It has been defined as follows: ing equations are the momentum and pressure Poisson equations, which have been defined as follows: where , , , and are the fluid's density, velocity, pressure, and viscosity; and is the turbulent viscosity. Furthermore, g is gravitational acceleration and is intermediate velocity.
This algorithm was developed to model special prob-lems with the effects of the free surface and forced motion of solid boundaries. To overcome SPH defects, some modifications are needed. The present algorithm contains the main body and different sub-algorithms, which has been adjusted according to the ISPH algorithm by . Other sub-algorithms added are: (1) Forced motion of structures; (2) Free surface detectors; (3) k-ε equations solver and turbulence viscosity calculators; (4) Shifting algorithms.
In the present algorithm, a predictor-corrector scheme has been implemented. In the first step, according to the gravitational and viscous terms of the momentum equations, the velocity is predicted as follows: where is the unit vector (from j-th particle to i-th particle), is the interpolation function, is the volume of j-th particle, and B is the kernel gradient corrective tensor, which has been proposed by Bonet and Lok (1999) as: ν t Thus, is the turbulent viscosity which is calculated by the following equation: where , is the particle spacing and is the strain rate of the mean flow (Aly and Lee, 2014;Shamsoddini and Aminizadeh, 2017). However, in the present study a k-ε SPH method has been used. According to k-ε method is calculated by: where k is calculated as follows (Violeau and Issa, 2007): . Its SPH discretization is: Then, is calculated using Eq. (9) and is compared with the calculated value using Eq. (8). By substituting into Eq. (6), intermediate velocity is calculated. After calculating the intermediate velocity, the pressure has been calculated using Eq. (5): In this study, the above equation has been explicitly solved as follows: ∇ · r ∇ · r ∇ · r = 2.0 ∇ · r If the i-th particle is on the free surface, then p i is set to equal zero. To detect free surface particles, a sub-algorithm was developed. For each particle, is calculated for each particle. For the two-dimensional case, should be equal to two ( ). SPH discretization of would be: ∇ · r ∇ · r < 1.6 However, would be smaller than 2 for the free surface particles. In the present stud, all particles with are considered as free surface particles. After calculating pressure, the velocity is corrected as: Another sub-algorithm is the definition of the force motion of solid bodies. In this section, the formulation of motion for solid particles is exerted. For all particles, the new position is calculated as: ∆r i Defects, tensile instability, and clustering distributions are SPH modeling complications. To prevent them, a shifting algorithm by Shamsoddini et al. (2014) was defined. First, is calculated as the shifting particle vector: ε δr where is a constant varying between 0 and 0.1 and is: r iri ∆r i Homogeneous distribution of the particles around the particle i leads to =0.0. If ≠0.0, then the particle is shifted by . Finally, it is necessary to correct the flow field variables in the new position. According to the first order Taylor series expansion, these corrections are as follows: To apply the no-slip condition, near each wall boundary, two rows of dummy particles have been arranged. The velocity of each dummy particle has been calculated using its corresponding position. If the wall has a linear motion or is fixed, the velocity of the dummy particle is the same as its corresponding wall particle. For rotational cases, the angular velocity of the dummy particles and wall particles are the same. The pressure equation for dummy particles is obtained by dot multiplying the normal vector of the surface (n w ) by the momentum equation: Because of the fixed position of dummy particles relative to the wall particles, the above equation can be discretized according to the finite difference method, as follows: where is the distance between dummy particles and its corresponding wall particles. Usually, the second term of the right-hand side of this equation is small enough and can be neglected. If wall velocity is constant, the equation is converted to Neumann conditions for pressure. The summary of the developed code has been presented in Table 1.
The presented algorithm in this study is suitable for modeling of fluid flows with the free surface engaged with the moving rigid bodies. In the following, the accuracy of the present algorithm has been evaluated using the experimental data. Therefore, after the validation test, the experimental setup has been explained, and then, the numerical and experimental results have been compared and discussed.

Validation test
The collapse of a water column in a tank due to the gravity is a traditional problem to test the codes developed for free surface flows. This problem is known as dam breaking. The geometrical properties of the problem have been shown in Fig. 1. Height of the water column is twice of its width. The gravity force causes that the fluid flows down and right.
As shown in Fig. 2, a continuous fluid flow is observed along the horizontal surface until the flow reaches the vertical surface. Then, the flow goes up along the right-hand side vertical wall. After that, the flow returns back and is accumulated. Fig. 3 shows a quantitative comparison among the results of the present code with the experimental results V * ,n+1 compute using Eq. (6); if i is an internal or wall particle then calculate pressure using Eq. (14); else if i is a dummy particle update pressure of dummy particles using Eq. (22); update concentration of dummy particles using Neumann condition; else if i is a free surface particles ( ); p i =0.0; end if; if i is a moving rigid body particle then impose formulation of motion; else update velocity using Eq. (16) end if update the position using Eq. (17); if i is an internal particle then  of Koshizuka and Oka (1996) and also numerical results of Violeau and Issa (2007). In this figure, the position of the leading edge of the fluid flow has been plotted. As depicted in this figure, the present ISPH k-ε results are closer to the experimental results than the WCSPH k-ε results.

Results and discussion
In the present study, the results of the numerical simulation are compared directly with the experimental results. The experimental setup consists of an electrical motor, a crank mechanism, and a glass box (14 cm×41 cm×20 cm) mounted on four wheels as shown in Fig. 4. First, the crank and the slider are connected to the engine, the glass box is set on a cart (four wheels), and the cart is attached to the crank. The engine is set to a certain rpm. The glass box is filled with water to the height of 2.5 cm. A CCD camera with 21MPixel is used to take the video films. Three cases are adjusted: (1) The box without the baffle; (2) The box with a baffle; (3) The box with two baffles. Fig. 4 shows the box with two baffles. The baffle height is 2.2 cm. To extract the images from the video films at a certain time, the "Video Image Master Pro" software is used.
The motion of liquid within the tanks under the vibra-tion condition is an interesting problem for engineers and scientists. Modeling this problem has useful results for predicting the sloshing phenomena or fluid motion in the containers or dams with shallow water under the earthquake or vibration conditions. In this phenomenon, the annoying forces and fluctuations may cause greater forces and momentum, which should be controlled. So, in the present study, to control the violence of water flow, the vertical baffles are considered and tested numerically and experimentally.
For the first investigation, the shallow water sloshing phenomenon is investigated. The tank motion is defined as: where A = 0.045 m and T = 1.0 s. The effect of the particle size on the accuracy has been investigated for four different particle sizes: δ=L/100, δ=L/200, δ=L/400, and δ=L/600. For these cases, the curve of free surface after a time period of motion has been plotted in Fig. 5. The convergence between the case with δ=L/400, and δ=L/600 is obviously observed. So, the case with δ=L/400 is selected for the SPH modeling in the present study. Fig. 6 shows both of the numerical and experimental results for the glass box with the horizontal oscillation with certain frequency and domain. This motion is transmitted to the water which made the liquid to flow right and left. A continuous fluid flow is observed along the horizontal surface until the liquid reaches the vertical wall. Then, the liquid goes up along the right-hand side vertical wall. After that, the liquid returns and is accumulated, as shown in Fig.  6. Under this condition, a violent flow is observed in all of the cases shown in Fig. 6. This flow frequently creates collapses of water on the vertical walls. This violent flow creates high amplitude forces with high wave height. If the wave height is controlled, the force domain is also controlled.
To reduce the wave height, a vertical baffle is tested which is set in the middle of the horizontal bottom plane of  Koshizuka and Oka (1996) and numerical results of Violeau and Issa (2007). Fig. 4. Experimental mechanism considered for investigation of the effects of baffles on the shallow water sloshing phenomenon.

Fig. 5.
Curve of free surface for four different particle sizes (the effect of particle size) after a time period of motion (t/T=1.0). the box. The results of this case are shown in Fig. 7. In this figure, the results of the numerical simulations are compared with the experimental results. As shown, there is a proper agreement between the numerical and experimental results. Also the comparison between the results shown in Fig. 6 and Fig. 7 demonstrates that the presence of baffle has a proper effect on the reduction of the wave height. As shown in Fig. 7, the height of collapse of water on the vertical walls is significantly reduced. Now, by increasing the number of baffles, it is expected that the maximum wave height has a more decrement. Therefore, two vertical baffles were set at each one-third of the bottom plane length. The results of the water sloshing in the presence of two vertical baffles are shown in Fig. 8. As depicted in this figure, the maximum height of the waves is considerably reduced in comparison with the cases of with one baffle and no baffle. In order to have a quantitative comparison of the three investigated cases, the time variation of the pressure at a point on the right-bottom corner of the box for these cases is plotted in Fig. 9. This figure shows that the pressure variation is a periodic function of time. The force sinusoidal motion of the tank is gradually transmitted to the fluid. A periodic flow is also created in the tank. As depicted, by increasing the number of baffles, the pressure variations decrease considerably. For the case with one baffle, the pressure variation domain is about 20% and for the case with two baffles, it has about 45% decrement in comparison with the case with no baffle.

Conclusion
In the present study, a robust modified explicit ISPH method was developed and applied for simulating the shallow water sloshing phenomena in a rectangular box. The method was improved by using the kernel gradient correction, free surface particle detector, k-ε turbulent model and particle shifting algorithm. Simultaneously, a mechanism was designed and constructed to record the shallow water sloshing details, experimentally. A comparison between the numerical and experimental results approved the accuracy of the present algorithm for modelling the sloshing phenomena. The main aim of this study was to investigate the effects of the baffle on the controlling of the shallow water sloshing phenomena. Therefore, the cases with one and two baffles were compared with the case with no baffle, numerically and experimentally. The results show that the presence of baffle has an essential role in the reduction of the sloshing fluctuations. It was shown that, utilizing one or two baffles in the box decreases the pressure fluctuations about