Optimal Design of Circular Baffles on Sloshing in a Rectangular Tank Horizontally Coupled by Structure

Parametric studies on the optimization of baffles on vibration suppression of partially filled tanks coupled by structure have been widely conducted in literature. However, few studies focus on the effect of the position of the baffles on fluid flow stratification and dampening the motion. In the present study, a numerical investigation, an engineering analysis, and optimal design study were performed to determine the effect of external flow on circular obstacle baffles performance on suppressing the vibrations of coupled structure in a closed basin. The single degree of freedom model (mass–spring–damper) is used to model the structure that holds the tank. The coupled system is released from an initial displacement without a velocity. The governing mass, turbulent Navier–Stokes momentum, volume of fluid, and one degree of freedom structure equations are solved by the Pressure-Implicit with Splitting of Operators algorithm in fluids and Newmark method in structure. Based on a detailed study of transient structure motion coupled with sloshing dynamics, the optimal baffle location was achieved. Optimal position of the baffle and its width are systematically obtained with reference to the quiescent free surface.

Based on the above literature search, there is an absence of study on the effect of pipe shaped baffles on the fluid flow in the rigid tank coupled with structure. The aim of the current study is to investigate the effect of the presence of pipe shaped baffles in the tank on the fluid flow around it and the effect of different immersing of the baffles as well as the liquid fill levels on the free surface profiles and vibration mitigating behavior of fluid sloshing on coupled motion.

Setup
Consider a partially filled tank in 2D rectangular shape as shown in Figure 1b with the maximum height of 300 mm and the length of 500 mm, respectively. For the basic case (without baffles), the filling ratio is considered to be 110 mm. The fluid density (ρ) is considered constant in all simulations. All numerical studies done in this research are summarized in Table 1. For the incompressible fluid, the governing equations are which is continuity equation adopted by ρ = const in ∂ρ ∂t + ∇ · (ρu) = 0 and which is the momentum equation adopted by Φ ≡ Φ + φ (for variables, Cartesian velocity components V i and pressure (P)) in ρ Du Dt = −∇p + µ∆u + ρf. In Equation (2), the generalized central second moment τ(V i , V j ) is the sub-grid stresses tensor related to strain-rate tensor where δ ij is the Kronecker delta, and ν is the turbulent viscosity (eddy) (nu = C µ k 2 if k-model is used for the turbulence) and turbulence kinetic energy (k) is found from The coefficients are the same as [74,83]. The k 3/2 / and Kolmogorov length scale (ν t +ν) 3 1/4 are two measures of the grid size. The method of the volume of fluid is used to model the free surface of the fluid [34,39]. The method computes the phase fraction α (α = 1 means pure liquid and α = 0 means pure gas since ∑ α i = 1) through the domain by solving the material derivative of α equals zero as The surface tension source term in Equation (2) (q by continuum surface force model) is found from where surface tension is σ = 0.076 N/m and κ is defined in terms of the divergence of the surface normal (κ = ∇ · n = ∇ · ∇α). The coupling equation between the fluid and solid shown in Figure 1b is The flow regime around pipes is detected based on some parameters such as bulk fluid turbulence intensity and aspect ratio. The most important parameter is the Reynolds number, where D is the pipe diameter, V ∞ is the time-averaged velocity, and ν is the effective fluid kinematic viscosity (see Table 2). Based on the data provided for set-up, the Reynolds number of the whole tank is in the transition in the boundary-layer flow regime while the Reynolds number of the pipe is in the transition in the free shear-layer flow regime around pipes.   The kinematic (no mass flux across the free surface) and dynamic (dynamic equilibrium of free surface) are free surface boundary conditions and wall boundary conditions are no-slip velocity, where n is the unit normal vectors to the walls.

Validation and Grid Independence
Pressure implicit method with splitting of operators [112] is used for the solution of the system of Partial Differential Equations (PDEs) appearing in Equations (1), (2), (5)- (7) and (9). The three-step method (predictor-corrector-regularization), first by a guessed pressure (∇p), finds turbulent velocity field via iterative resolution techniques (14) and then corrects pressure, and finds corresponding velocity components (satisfy ∇ · u = 0) with the Poisson equation: Finally, in the regularization step, by putting the difference of the corrected and guessed pressure values in the momentum equation, correction of the velocity can be determined.
The problem shown in Figure 1b with parameters of Table 3 is solved by various mesh sizes. The approach outlined above would be true if it has small error for the given mesh and one can make sure that the solution is also independent of the mesh resolution. The convergence based on average of the Root Mean Square (RMS) Error values over the domain compared with 10 5 nodes was performed. The 10 5 nodes solution is used as the smallest mesh that gives the mesh independent solution. The results are summarized in Figure 2a. Because the solution is not changing significantly with the refinement of mesh from 10 4 nodes to 10 5 nodes, we have achieved a mesh independent solution. You need to refine the mesh more, and repeat the process until you have a solution that is independent of the mesh. You should always do this (to reduce your simulation run time). The vessel motion is X s , which is defined as in [35]. The outcomes appeared in Figure 2a shows that the numerical simulation here has a good agreement with the Finite Element Method (FEM) benchmark results. The code will develop quickly with high precision when Courant number is equivalent to 0.5, speaking to a generally extensive time step. Figure 2b is the grid accuracy check of work. A reasonable accuracy can be seen between 8 K work and 10 K work in the calculation of structure location under the same conditions, while the contrast between 10 K work and 12 K work can be disregarded. With respect to utilization of calculation assets, a 10 K element number with Co = 0.5 is used for other cases. In this manner, all recreations worked and there is an element number of around 10 K. Convergence for a Steady State simulation satisfies the following three conditions: Residual RMS Error values have reduced to an acceptable value (typically 10 −4 ), monitor points for our values of interest have reached a steady solution, and the domain has imbalances of less than one percent. Here, the converging criteria for fierce continuity equation is 10 −7 , for momentum equation is 10 −7 , and for turbulence equations (dissipation and energy rates) are 10 −5 . In turbulent flows, there is a considerable challenge in error control, especially of the near-wall boundary condition. As the wall-functions are used in the near-wall boundary layer, the size of the near-wall cell is limited [113]. The low Reynolds model causes a very large number of nodes at the near wall boundary layer. Other than spatial error, in transient turbulent flow calculations, the temporal errors appeared. The temporal error of each step is proportional to ∆t 2 other than a Courant-Friedrick limit (CFL) used for numerical stability for the time advancement [114].

Effect of Gap with the Bottom Wall
Numerical recreations are performed for three unique places of the ring perplex with reference to the free surface. Adequacy in controlling the free surface variety under a remotely forced horizontal excitation is attempted. Free surface elevation of the fluid at various distances from the side wall is presented in Figure 3. Figure 3a presented the fluid level at t = 0.3 s for Gap ratio equals to 0.1 which the free surface doesn't affect properly by baffle. Figure 3b presented the fluid level at t = 0.45 s for Gap ratio equal to 0.6. Figure 3c presented the fluid level at t = 0.2 s for Gap ratio equal to 0.8 as shown in the approach of baffle to free surface causing higher wave breaking. Based on the analysis of Dodge et al. [23], which showed that the interface shape is the most important parameter, we started here with the effect of gap with the bottom wall. Consider a cylindrical baffle that used in the simulation as shown in Figure 4, with fill level = 37%. The liquid volumes in all cases are equal. Figure 4 shows the pressure distribution for filling level of 11 cm and 6.6 cm distance of 3.85 cm from the bottom at a time of 0.1 s, which illustrates the effect of initial acceleration on the curvature of interface. Fluid flow advection mainly happens between baffle and the bottom wall.  Pressure and velocity in liquid for filling level = 37% and 0.6 × h f distance from bottom at t = 1 × (s) appear in Figure 5. Velocity near the cylinder in the fluid stage is higher than the other points. As is shown, the existence of the baffle causes fluid flow stratification. Due to the tuned liquid damper, the zone of fluid wet divider is bigger; accordingly, it sets aside more opportunity for the viscous effect to be exchanged to the bulk fluid.  By using the tool of dimensional analysis, we can classify the regime of fluid-solid interactions. By defining a dimensional velocity by dividing U s by U f , one can define the dimensional number. Reduced velocity number, U R , is a dimensionless number that shows that the relative velocity of fluid to solid is defined as where h f in this case stands for fluid height, g is the gravity coefficient, x 0 is the initial location of structure from static equilibrium point, and w s is the liquid angular velocity. As the baffle is considered as a rigid body, Mass number (relative density), Cauchy number (fluid pressure to Young modulus) will not affect the system, while the Reynolds number (viscosity effect around the cylinder) and Fraud number (gravity pressure effect) are important. The added mass is constant in all U R , the added damping is ∝ U R , and the added stiffness is ∝ U 2 R . As here the T f ≈ T s or U R ≈ 1, the fluid motionp = −ω 2 f p and solid motionẍ s = −ω 2 s x s are coupled via dynamic condition Average distortion versus gap distance ratios are shown in Figure 7. Although all the cases shown in Figure 6 have a lower value of X s than the free vibration of the structure, the total behavior is investigated in Figure 7. As is shown, the average of norm of structure position in case of h c = 0.8h f is higher than the free vibration of structure because of the transient overshoot of this case in the first period of vibration. In addition, the best cases happen in h c = 0.5h f , which can be considered as the optimal location. When the baffle is at h c = 0.5h f , the advection has a wide range, and the average magnitude of velocity is higher. Although the maximum displacement of the tank solid is different in these two configurations, the velocity field in the middle and bottom parts of the tank is similar.

Effect of Filling Level
The effect of filling ratio is shown in Figure 8. The solid displacement versus fill level ought to have a pattern as the dashed line in the tank without a perplex. Normally, the increase of filling ratio helps increment the fluid frequency, advance dissipation and bring damping up in the tank. Furthermore, the sloshing motion apportioning model states that the dissipation from baffle to tank is comprised of three regions: the damping of the top breaking wave, the dissipation for exposing wave on the sides and the dissipation from the flow in the bottom of the cylinder. For the current situation, dissipation exchanged from the fluid wet boundary affects the adjacent fluid. Subsequently, the wet region near the baffle and thickness of the boundary layer of fluid close baffle, which compare to the capacity of all boundaries of the vessel that exchange momentum by fluid, impact dissipation incredibly. On account of the presence of cylindrical baffle near the surface, when fill level is near the wall of the cylinder, fluid will move up to the top boundary of the cylinder by surface wave; along these lines, dissipation exchanged to the fluid stage. In this manner, the increase in dissipation will be smaller in the event that it is working. Thus, we get a sharp decrease as fill level increases. The progress point is somewhat higher than 40%, which can submerge the base baffle in an ordinary system, at around 50%. For the current paper where the D = h f 2 at the point when fill level is 18%, the fluid is enough to fill the space between the cylinder and the bottom wall, and fluid that filled later will show up at the center of the tank, an area that isn't viable for blocking the fluid between two regions. In this way, for levels somewhat over 18%, interfaces are comparatively close dividers, as shown in Figure 9, so the damping ratio is increased. In addition, when fill level is low, 30% for instance, fluid predominantly exists under the cylinder base flow and isn't sufficient to cover the divider between two zones. At the point when the fill level is over 70%, the gas momentum is huge in the upper segment over the cylinder; in this way, the impact of baffle is feeble. Consequently, the baffles can moderate momentum ascension for fill levels in the cylinder and the bottom, which is useful in light of the fact that the liquid in the rigid tank normally lessens from full to purge during the mission. Average distortion versus gap distance ratios is shown in Figure 8. Although all the cases shown in Figure 9 have lower values of X s than the free vibration of structure, the total behavior is investigated in Figure 8. Free surface elevation of the fluid at various filling levels is shown in Figure 10 for filling ratio of 14% in Figure 10a and for filling ratio of 42% in Figure 10b, respectively, for time equal to 2 sonds. The free surface is not smooth here as the wave breaking happens through the sloshing. As is shown, lower filling ratio causes higher wave breaking and thus lower displacement for the structure, as shown in Figure 9. In addition, the difference between 11 cm and 7 cm levels is sensitive to the forces in the vehicle.

Distance from the Side Wall
Control of slosh powers and free surface wave stature affected by horizontal excitation is absolutely alluring. Albeit a few dynamic and aloof techniques have been proposed in the open writing, utilizing inside puzzles is the most straightforward of every such arrangement. For a straightforward stockpiling tank, a position with reference to the free surface, width and thickness of the bewilder is a basic plan included. Simulations are performed for tanks with different baffle distances from the left wall in Figure 11. During the simulation, fluid interface climbs up to the edge of tanks at the beginning, then covers baffle as much as it can by surface tension, and then becomes damped for the rest of the simulation. The effect of distance from the side wall on liquid sloshing could be considerable when the tank is subjected to horizontal excitation. Both low and extensive separations between wall and baffle can moderate the fluid motion, and the decrease of solid motion ascending against distance is shown in Figure 11 with fill level of 50%. This is on account of the fact that, at zero distance, which isn't exceptional in moderating sloshing, the liquid changes dramatically, and thus the fluid vortexes as well. According to the normal VOF technique, fluid volumes with close interfaces have decimal qualities. The adjustment in fluid volumes results in development of an interface. During the damping procedure in the first initial second, interface in the baffled tank encounters quick acceleration. Be that as it may, as separation builds, the vacillation is essentially facilitated. We can see little changes in the distance ratio = 1 3 case; however, for distance ratio = 1 8 , the fluid volume is just about consistent near 1. As distance increases, the contribution of reducing sloshing waves changes from diffusion to breaking waves as the wet area increases. However, in cases of distance ratios 1 8 and 1 3 , the sloshing phenomenon is weakened by baffles, but the liquid near the wall is too thin to weaken the wave momentum effectively as the liquid is fluctuating.
(a) Distance ratio = 1 3 (b) Distance ratio = 1 8 Figure 11. Free surface elevation of the fluid at various distance from the side wall.
The comparison of solid distribution and velocity field between distance ratio is 1 8 and 1 3 is shown in Figure 11. Thus, we get the profiles as shown in Figure 12. As is shown in Figure 12a, a periodical force is applied to the tank wall. The differences in 17 cm and 21 cm distances are quite small such that one that cannot be correctly predicted will cause the failure of the mission. As is shown in Figure 12b, a large distance from the wall is preferred.

Conclusions
In this paper, a parameter study on the optimization of baffle on vibration suppression of a partially filled rectangular rigid tank coupled by structure has been performed. The coupled system is released from an initial displacement without a velocity. Navier-Stokes governing equations coupled by the equation of volume of fluid and a single degree of freedom structure are solved. As is shown, higher performance can be easily achieved by considering effective parameters. Specific findings from the present numerical simulations can be outlined as follows: • The near wall zone plays an important role in dampening the fluid momentum in the tank.

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The baffle located close to the free surface is effective in control of surface fluctuations, as well as concomitant force and pressure.

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The filling ratio affects the wet wall fraction and sloshing frequency.

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The existence of distances from the wall influences the wet wall area (surface tension effect), and liquid can climb up to the baffle if the baffle is near the surface.

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As the sloshing in the tank is steady, baffles can do little with wave breaking and have no influence on the behavior of liquid.

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The liquid thickness has more importance in slosh damping than wet area.

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The parameter studied here based on importance are the distances, filling ratio, and gaps of the baffle • Based on a detailed study of transient structure motion coupled with sloshing dynamics, and by optimizing the parameters of baffle, optimal baffle location was achieved.