Analysis of Inherent Frequencies to Mitigate Liquid Sloshing in Overhead Double-Baffle Damper

Abstract

A disco-rectangular volume-fraction-of-fluid (VOF) model tank of a prismatic size is considered here for investigating the severe effect of overhead baffles made of three different materials, nylon, polyamide, and polylactic acid. In this work, the overdamped, undamped, and nominal damped motion of baffles and their effect are studied. In this research, the behaviour of different material baffles based on the sloshing effect and natural frequency of each baffle excited in damped, undamped, and overdamped cases is studied. VOF modelling is carried out in moving Yeoh model mesh with fluid–structure interaction between the water models for various baffle plates. The results of the water volume distribution and baffle displacement operating between a sloshing time of 0 and 1 s are recorded. Also, a strong investigation is carried out for the water volume suspended on overhead baffles by three different material selections.

1. Introduction

Liquid sloshing in rectangular tanks with vertical baffles has been extensively investigated in the literature, refer to the studies [1,2,3,4,5] and the references provided therein. Several methods have been used to study the effect of vertical baffles on sloshing behaviour using a semi-analytic approach to study liquid sloshing in a rectangular tank with dual elastic vertical baffles and walls, refer to [6]. This method entails finding the formal solution of the potential function in each sub-domain using the separation of variables method, then deriving the baffle or wall deflection solution from the transverse vibration equation. The Eigen equation is established using free surface, interface, and coupled vibration conditions, and the coupled frequency is calculated by solving this equation. The coupled dynamic response equation is then derived from the free liquid surface equation, and the coupled vibration equation is applied see [7]. Porous media theory was used to simulate porous baffles and investigate their effects on the sloshing flow. According to the article, the wave-damping performance of porous baffles is poor when the excitation frequency approaches the first-order natural frequency. This unexpected finding shows that a porous baffle may not be helpful in preventing sloshing in some frequency ranges. It was discovered that increasing the height of the baffle and decreasing the porosity coefficient of the baffle can improve its wave-damping performance. This result is surprising as it indicates that modifying the physical characteristics of a porous baffle can significantly improve its effectiveness in reducing sloshing.

Computational fluid dynamics simulations were conducted to analyse the slosh response of a rectangular tank equipped with perforated baffle plates [8]. Future research could focus on the application of different types of baffle plates, such as solid or perforated plates, to compare their effectiveness in slosh damping under pitch excitation. Additionally, investigating the slosh response under different types of excitations, such as surge or roll excitation, would provide a more comprehensive understanding of slosh-damping behaviour in rectangular tanks.

Research was proposed using a linearised analysis model in [9] to estimate the damping ratio of a rectangular tank with bottom-mounted vertical baffles. The paper provides a linearised analysis approach that uses potential flow and linear wave theory to determine the damping ratio of a rectangular water tank with bottom-mounted vertical baffles. The model takes into account the hydrodynamic interaction of the vertical baffles and fluid velocity, as well as the extra mass coefficient. Shaking table tests were performed on scaled replicas of the rectangular tank with bottom-mounted vertical baffles to ensure the suggested analytical model’s accuracy. The article also conducts parametric experiments to explore the impact of numerous parameters, such as the distance between vertical baffles, water depth ratios, excitation amplitudes, and the number of bottom-mounted vertical baffles, on the damping ratio.

A study was conducted using a 3D simulation to study the efficiency of horizontal and vertical baffles in decreasing sloshing in a prismatic tank [10]. The paper contributes to the understanding of sloshing phenomena in prismatic tanks and provides insights into the effectiveness of baffles in mitigating sloshing motion. It introduces a novel methodology based on potential flow and linear wave theory to estimate the damping ratio of rectangular water tanks with bottom-mounted vertical baffles, taking into account the hydrodynamic interaction and natural frequency reduction effects. This study provides valuable insights into the liquid-sloshing effect in rectangular tanks with vertical baffles and offers guidance for designing effective anti-sloshing measures.

One notable weakness in the approaches used in these papers is that they mainly focus on numerical simulations and analytical models rather than experimental validation. While some papers mention the use of shaking table tests to validate their models, the overall emphasis is on theoretical analysis and computational fluid dynamics (CFD) simulations.

Additionally, the papers do not provide a comprehensive comparison of different anti-sloshing techniques or approaches. Each paper focuses on a specific aspect, such as the use of vertical and horizontal baffles, bottom-mounted vertical baffles, or perforated baffle plates. Therefore, a holistic evaluation of the strengths and weaknesses of different approaches is lacking.

It is worth noting that the papers mainly focus on rectangular tanks and do not extensively explore the applicability of the proposed approaches to other tank geometries or shapes. The studies primarily investigate sloshing in 2D or prismatic tanks, which may limit the generalizability of these findings to other tank configurations.

Various parametric studies investigate the effect of various factors, such as the damping ratio, on the distance between vertical baffles, water depth ratios, excitation amplitudes, and the number of vertical baffles positioned at the bottom. The proposed methodology efficiently predicts the damping ratio by incorporating the hydrodynamic interaction and natural frequency reduction impact caused by the installed bottom-mounted vertical baffles [11,12].

The proposed semi-analytical method can be extended to study the dynamic response of tanks with different geometries and configurations, such as tanks with non-rectangular shapes or multiple baffles. It would be interesting to explore the influence of different excitation types, such as harmonic and seismic pitching excitations, on liquid sloshing behaviour in baffled tanks [13].

A few studies contribute to the field by providing a comprehensive analysis of the sloshing phenomenon in different types of liquid storage tanks, including cylindrical and bi-lobed tanks, with and without baffles. The influence of baffle parameters on sloshing dynamics has been evaluated, providing insights into the design of tanks to minimise the horizontal forces exerted during tank motion [14].

One study utilises innovative semi-analytical methods, such as the scaled boundary finite element technique (SBFEM), spectral meshless radial point interpolation method (SMRPI), and singular boundary method (SBM), to study the effects of perforated and imperforated baffles on sloshing pressure. The study uses a second-order time discretization approach to increase the numerical accuracy of predictions and compare the results with experimental data [15].

Some studies do not consider the effects of other factors, such as fluid viscosity, tank material properties, and external forces, which could influence sloshing behaviour in real-world applications, for example see [16].

Another study investigates the effects of the different configurations of baffles on sloshing dynamics, providing valuable information on the spilling out of a liquid and the amount of liquid left in the tank after a particular decisive time [17].

A scientist-conducted experimental study relies on the synchronised measurements of dynamic pressure on the tank walls and sound pressure level but does not provide detailed information on the measurement techniques or potential sources of measurement errors [18].

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Some details needed to explain the potential limitations and assumptions in the numerical simulations and analytical methods used are not explicitly discussed [19];

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The numerical simulations in Ref. [20] were based on the finite element method and arbitrary Lagrangian–Eulerian (ALE) method, which allowed a comprehensive investigation of sloshing dynamics in rectangular tanks with impermeable and permeable baffles;

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Ref. [21] utilised the Compressible Moving Particle Explicit (WCMPE) method–conversion of simulated data and images for studying sloshing behaviour weekly. They utilised the Navier–Stokes solver for numerical investigations and studied insights on the parametric identification of wave behaviour and hydrodynamic load;

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Ref. [22] conduct numerical study using the Moving Particles Semi-Implicit (MPS) method and studied analytical estimations for the natural frequencies of floating baffles.

This research concentrates on the application of baffles made of a suitable material possessing a good VOF model to enable one-third of the fluid volume inside the tank to slosh across the entire disco-rectangular tank. Three different arbitrary materials—nylon, polyamide, and polylactic acid (PLA)—possessing a low-viscosity smooth surface and low density (thin sheets) were selected. The following research describes the fluid–structure interaction of two overhead baffles made of different materials in a disco-rectangular tank and predicts the natural frequency of the baffles, volume distribution of the fluid sloshes, and directional displacement of the baffles during sloshing.

2. Geometrical Arrangement of FSI Model

A tank with a disco-rectangular shape was used, consisting of a box width of Wb, box length of Lb, box height of Hb, and a semicircle with a diameter equal to the box height, as shown in

Table 1. Geometrical arrangement of FSI model.

Description	Dimension	Geometry Description
Hb	0.01 m	Box height
Wb	0.005 m	Box width
Lb	0.03 m	Box length
Xo	0.015 m	Position of obstacle
Do	0.0003 m	Baffle thickness
Ho	0.009 m	Baffle height
Wo	0.004 m	Baffle width
Xi	0.01 m	Initial position of the interface

. Two overhead baffles were positioned in between the two interfaces positioned at Xi. The distance between the two baffles was similar to the initial position of the interface, and the first baffle was fixed 0.002 m from the position of Xi. The dimensions of the baffles, baffle thickness (Do), baffle height (Ho), and baffle width (Wo) are shown in Table 1.

All dimensions were plotted in the work plane geometry and extruded in the modelling of the disco-rectangular tank. The other dimensions of the baffles were plotted in the work plane geometry with the known locations and extruded in the modelling of the overhead baffles positioned at 1 and 2, as shown in Figure 1.

3. Moving Mesh Modelling

In domains with a free displacement of baffles, the deforming domain (fluid–structure interaction bodies called baffles) and free-feature working fluid in the moving mesh and deformed geometry interface (the fluid–structure interaction between baffles and fluid) are solved by the mesh displacement equation. Given the constraints applied to the boundaries of the mesh, the deformation equation smoothly deforms the mesh. Appropriate mesh smoothening should be selected from Laplace smoothing, Winslow smoothing, hyperelastic smoothing, or Yeoh smoothing.

To set the mesh smoothing methods, the mesh smoothing type list selected in the deforming domain node of the moving mesh or deformed geometry should be used as a reference. To observe these mesh-smoothing approaches, the spatial frame’s u and v coordinates, as well as the material frame’s U and V reference coordinates, should be considered.

If Laplace smoothing is used, the program adds distorted mesh coordinates u and v as degrees of freedom to the model. In static cases, it solves the following equation:

$${\displaystyle \frac{{\partial }^{2}u}{\partial U^2}}+{\displaystyle \frac{{\partial }^{2}v}{\partial V^2}}=0$$

(1)

and in unsteady cases, it solves the equation

$${\displaystyle \frac{{\partial }^2}{\partial U^2}}{\displaystyle \frac{\partial u}{\partial t}}+{\displaystyle \frac{{\partial }^2}{\partial V^2}}{\displaystyle \frac{\partial v}{\partial t}}=0$$

(2)

Similar equations apply to the v coordinate.

If Winslow smoothing is used, the software solves the problem

$${\displaystyle \frac{{\partial }^{2}U}{\partial u^2}}+{\displaystyle \frac{{\partial }^{2}U}{\partial v^2}}=0$$

(3)

with U and V both satisfying Laplace equations as functions of the u and v coordinates.

The hyperelastic mesh smoothing method, inspired by neo-Hookean materials, seeks a minimum of mesh deformation energy

$$W={\int }_{\Omega }{\displaystyle \frac{\mu }{2}}\left(I_1-3\right)+{\displaystyle \frac{\kappa }{2}}(J-1{)}^{2}dV$$

(4)

where μ and κ are the artificial shear and bulk moduli of the baffles, respectively, and the invariants J and I₁ are given by

$$J=\mathrm{Det}\left({\nabla }_{U}u\right)$$

(5)

$$I_1=J^{-2/3}\mathrm{tr}\left({\left({\nabla }_{U}u\right)}^T{\nabla }_{U}u\right)$$

(6)

The Yeoh mesh smoothing approach draws inspiration from hyperelastic materials, specifically the three-term Yeoh hyperelastic model, which is a generalisation of a neo-Hookean material. It utilises the strain energy of the shape.

$$W={\displaystyle \frac{1}{2}}{\int }_{\Omega }{K}_1\left(I_1-3\right)+K_2{\left(I_1-3\right)}^2+K_3{\left(I_1-3\right)}^3+\kappa (J-1{)}^{2}dV$$

(7)

As previously stated, κ represents an artificial bulk modulus, whereas K₁, K₂, and K₃ represent further artificial material qualities. The default values for K₁ and K₃ are 1 and 0, respectively, and can only be altered in the equation of a free deformation domain. The value of K₂ determines the nonlinear stiffening of the artificial material selected for the baffle plate during deformation. It is specified as 10.

A symmetry/roller should be used on the flat boundaries of deforming domains that are also symmetry planes in the model or anywhere to keep boundary regions away from the migrating of the plane. This permits the mesh to move tangentially to the symmetry/roller plane but not in the usual fluid flow direction. The condition is that the normal mesh displacement in the prescribed normal mesh displacement is specified as zero. A symmetry/roller for the fluid domain of the disco-rectangular tank’s geometrical model exactly at the deforming domain of the middle interface volume containing the overhead baffles’ exterior planes should be selected.

The prescribed normal mesh displacement settings are used to specify the boundary’s displacement in the reference normal. The edges of the deforming domains should be marked. There are no limits on the tangential displacement. A free tetrahedral mesh is added to generate an unstructured tetrahedral mesh. This feature generates a mesh using the complementary domains, borders, edges, and points. The mesh solver may adjust the quantity, size, and distribution of elements using size and distribution subnodes, as shown in Figure 2.

Mesh Convergence Test

Four different categories of mesh were considered for the disco-rectangular tank: (a) Mesh 1 describes the normal physics-controlled mesh, (b) Mesh 2 describes the coarse mesh, (c) Mesh 3 describes the coarser mesh, and (d) Mesh 4 describes the fine mesh. The configurations of each mesh category are described in

Table 2. Mesh configurations for different categories of mesh.

Mesh Parameters	Mesh 1	Mesh 2	Mesh 3	Mesh 4
Minimum element size	0.2 mm	0.3 mm	0.4 mm	0.1 mm
Curvature factor	0.6 mm	0.7 mm	0.8 mm	0.5 mm
Maximum element size	0.67 mm	1 mm	1.3 mm	0.53 mm
Resolution of narrow region	0.7 mm	0.6 mm	0.5 mm	0.8 mm
Maximum element growth rate	1.15 mm	1.2 mm	1.25 mm	1.13 mm

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A parametric sweep can include multiple independent parameters directly for a fully potential disco-rectangular domain. More than one parametric sweep node can be added to create nested parametric sweeps by selecting mesh parameters, such as the maximum element size, minimum element size, maximum element growth rate, curvature factor, and resolution of narrow regions. The program then treats the parametric sweeps as a “nested for-loop” and indicates the nested structure using indentations of the sweep nodes’ names as Mesh 1, Mesh 2, Mesh 3, and Mesh 4.

The parametric sweep solutions were calculated for different categories of mesh configurations, and the displacement of baffles in the x-axis and the z-axis were plotted with respect to the fluid-sloshing time. The displacement of baffles in the x-axis vs. the fluid-sloshing time for different mesh categories was plotted and is shown in Figure 3. The displacement of baffles in the x-axis was found to be at its maximum for Mesh 4, and the amplitudes were neutralised in a sloshing time of 0.96 s, whereas minimum displacement was found for Mesh 2, and the amplitudes were not neutralised with respect to the sloshing time until 1 s. The displacement of the rest of the mesh configuration baffles in the x-axis was plotted in the moderate displacement field and not neutralised. Hence, Mesh 4 was considered for simplifying the numerical fluid flow problem. The displacement of baffles in the z-axis vs. the fluid-sloshing time for different mesh categories was plotted and is shown in Figure 4. From Figure 4, Mesh 4 yields the maximum displacement, whereas Mesh 2 yields the minimum displacement, but the displacement fields are damped and neutralised from a sloshing time of 0.125 s onwards. The rest of the mesh configurations yield moderate displacement and damping, and neutralization does not occur until a sloshing time of 1 s.

4. Physics Model of FSI

4.1. The Laminar Flow Interface

The laminar flow interface was used to calculate the velocity and pressure fields of a sloshing fluid. A flow is laminar as long as the Reynolds number is less than a specified critical value. At higher Reynolds numbers, disruptions tend to develop, causing the transition to turbulence. The physics interface is selected with flow boundary aspects, such as incompressible flow, weakly compressible flow (the density varies with temperature but not with pressure), and compressible flow at low Mach numbers (usually less than 0.3). It also enables the flow of non-Newtonian fluids. The laminar flow interface solves two equations: the Navier–Stokes equation for momentum conservation and the continuity equation for mass conservation.

The laminar flow interface can be used for both stationary solvers and time-dependent solvers. A time-dependent solver should be used in a high Reynolds number zone because these flows are intrinsically unstable. When the laminar flow interface is added, the Model Builder includes the following settings: fluid properties, wall (with the default boundary condition of no slip), and the initial values of the parameters selected for the fluid domain.

4.2. Phase-Field Setup

The phase initialization study step was used; however, two initial value nodes were required for the initialization to perform properly. As indicated in Figure 5, one of the initial value nodes should use water as the specified phase Liquid 1, while the other should use air as the specified phase Liquid 2. The initial fluid–fluid interface is then automatically applied to all internal borders between the two phases. When the level-set variable or phase-field variable is set to arbitrary, user-specified initial values must be selected. In this situation, smooth initialization was not used, and a transient time-dependent study with phase initialization was used instead.

Free energy is a function of the dimensionless phase-field parameter $\phi$: 

$$F(\varphi ,\nabla \varphi ,T)=\int \left({\displaystyle \frac{1}{2}}{\epsilon }^2|\nabla \varphi {|}^2+f(\varphi ,T)\right)dV=\int f_{\mathrm{tot} }dV$$

(8)

where ε is a measure of fluid-to-fluid interface thickness. The following equation describes the existence of the phase-field parameter:

$${\displaystyle \frac{\partial \varphi }{\partial t}}+\left(\mathrm{u}\cdot \nabla \right)\varphi =\nabla \cdot \gamma \nabla \left({\displaystyle \frac{\partial f_{\mathrm{tot} }}{\partial \varphi }}-\nabla \cdot {\displaystyle \frac{\partial f_{\mathrm{tot} }}{\partial \nabla \varphi }}\right)$$

(9)

where ftot is the system’s total free-energy density, and u is the velocity field for advection. The right side of the equation tries to minimise total free energy with a relaxation time determined by mobility γ.

The free-energy density of the ideal temperature of two immiscible fluids is equal to the sum of mixed and elastic energies. Mixed energy takes the Ginzburg–Landau form:

$$f_{\mathrm{m}\mathrm{i}\mathrm{x}}(\varphi ,\nabla \varphi )={\displaystyle \frac{1}{2}}\lambda |\nabla \varphi {|}^2+{\displaystyle \frac{\lambda }{4{\epsilon }^2}}{\left({\varphi }^2-1\right)}^2$$

(10)

The dimensionless phase-field variable ϕ is defined so that the volume fractions of the fluid’s components are (1 + ϕ)/2 and (1 − ϕ)/2. The mixing-energy density, denoted as λ (SI unit: N), and the capillary width, ε (SI unit: m), scale with interface thickness. The equation relates these two phase-field components to the surface tension coefficient is given by

$$\sigma ={\displaystyle \frac{2\sqrt{2}}{3}}{\displaystyle \frac{\lambda }{\epsilon }}$$

(11)

The PDE governing the phase-field variable is the Cahn–Hilliard equation:

$${\displaystyle \frac{\partial \dot{\varphi }}{\partial t}}+\mathrm{u}\cdot \nabla \varphi =\nabla \cdot \gamma \nabla G$$

(12)

G is the chemical potential, while γ is mobility. The Cahn–Hilliard diffusion’s time scale is determined by its mobility, which must be large enough to maintain a constant interfacial thickness while remaining small enough to prevent the convective effects from being unduly dampened. Mobility is determined by the tuning parameter γ = $\chi {\epsilon }^2$, which is a function of contact thickness. The chemical potential is:

$$G=\lambda \left[-{\nabla }^2\varphi +{\displaystyle \frac{\varphi \left({\varphi }^2-1\right)}{{\epsilon }^2}}\right]$$

(13)

Except for a thin region at the fluid–fluid interface, the Cahn–Hilliard equation requires ϕ to be either 1 or −1. The phase-field fluid–fluid interface separates the above equation into two second-order PDEs:

$${\displaystyle \frac{\partial \varphi }{\partial t}}+\mathrm{u}\cdot \nabla \varphi =\nabla \cdot {\displaystyle \frac{\gamma \lambda }{{\epsilon }^2}}\nabla \psi$$

(14)

$$\psi =-\nabla \cdot {\epsilon }^2\nabla \varphi +\left({\varphi }^2-1\right)\varphi$$

(15)

The following phase-field variable approximations were limited to the constraint requirements of the disco-rectangular model and are given by:

$${ϵ}_{\mathrm{p}\mathrm{f}}=1\times {10}^{-6} (Default interface thickness)$$

(16)

Mobility tuning parameter, χ = 5.

Phi derivative of external free energy:

$${\displaystyle \frac{\partial f}{\partial \varphi }}=0$$

(17)

A Wetted Wall is a unique feature that replaces the wall feature in the laminar or turbulent flow interfaces, as well as the No Flow function in the level-set interface. It is suitable for both laminar and turbulent flow, with wall functions or automatic wall treatment. The Wetted Wall boundary condition is appropriate for walls in contact with the fluid–fluid interface, as illustrated in Figure 6. This boundary condition allows fluid–fluid contact to travel along the wall.

For laminar flow, the wall boundary condition enforces the no-transmission condition u⋅n_wall = 0 and adds tangential stress on the form ${\mathbf{K}}_{\mathrm{n}\mathrm{t}}=-{\displaystyle \frac{\mu }{\beta }}\mathrm{u}$, where K represents the viscous stress tensor, and β represents the slip length. For numerical calculations, β = h is an appropriate selection for the mesh element size. The wall boundary does not reduce the tangential velocity component to zero. However, the projected tangential velocity component is zero at a distance of β from the wall. The specified contact angle is ${\theta }_w={\displaystyle \frac{\pi }{2}}$.

4.3. Shell Modelling

The shell interface, which appears under the solid mechanics branch when adding a physics interface, is used to simulate structural shells on 3D boundaries. Shells are thin, flat or curved structures with substantial rigidity. The interface employs MITC shell elements capable of analysing both thin (Kirchhoff theory) and thick (Mindlin theory) shells. Linear elastic material was selected from three different categories: (a) Material 1—nylon, (b) Material 2—polyamide, and (c) Material 3—polylactic acid. A linear elastic equation for displacements and the elastic material parameters are illustrated in

Table 3. Linear elastic material properties for different material selections.

Material	Material Properties	Value
Nylon	Young’s modulus	2.7 GPa
	Poisson ratio	0.4
Polyamide	Young’s modulus	2.9 GPa
	Poisson ratio	0.35
PLA	Young’s modulus	10.1 GPa
	Poisson ratio	0.33

. This material model assumes that the material is uniform across the thickness of the shell. A fixed wall constraint boundary condition for the overhead baffles was selected, as shown in Figure 7.

4.4. Multiphysics Solver Settings

Set the material properties to locally defined and choose the averaging method for density and dynamic viscosity in this area. Select the method for density and viscosity averaging. For both parameters, the default approach is Volume Average. In addition to the default approach, density averaging can be changed to the Heaviside function or Harmonic volume average and viscosity averaging to the Heaviside function, Harmonic volume average, mass average, or Harmonic mass average. When using the Heaviside function, enter a value for the corresponding mixing parameter, lρ or lμ. When the surface tension force is included in the momentum equation, check the shift of the surface tension force to the heaviest phase box. This can prevent major spurious oscillations in the velocity field for the lighter phase when there is a large density difference between the two phases. The degree of shifting is determined by the smoothing factor d_s,Fst (default 0.1).

The surface tension force is amplified by

$$f_s={\displaystyle \frac{2}{\left({\rho }_1+{\rho }_2\right)}}\left({\rho }_1\mathrm{H}\left({\displaystyle \frac{V_{f,1}-0.5}{d_{s,Fst}}}\right)+{\rho }_2\mathrm{H}\left({\displaystyle \frac{V_{f,2}-0.5}{d_{s,Fst}}}\right)\right)$$

(18)

Shifting surface tension force to the heaviest phase works best when paired with the Heaviside function to average density and viscosity. The two-phase flow and phase-field coupling features define the density and dynamic viscosity of the fluid used in the laminar flow interfaces, as well as the surface tension on the fluid–fluid interface as a volume force in the momentum equation. Temperature and/or pressure can affect a material’s density, which is automatically substituted by pref and Tref for incompressible flows (as given by the fluid–flow interface’s compressibility option).

When adding a physics interface, use the fluid–structure interaction (fsi) interface, which is located in the fluid flow branch. This interface allows you to model situations where a fluid and a deformable solid interact. The physics interface represents both the fluid and solid domains (structure), as well as a preset condition for interaction at fluid–solid boundaries. An ALE formulation is utilised to incorporate the geometrical changes in the fluid domain. The fluid might be either compressible or non-compressible. The flow regime can be either laminar or turbulent (assuming you have a CFD Module licence). The solid domain provides the same options as a solid mechanics interface, including contact conditions and nonlinear materials if the Nonlinear Structural Materials Module or Geomechanics Module is present.

The fluid–structure interaction interface simulates two-way coupling between solids and fluids through stationary or time-dependent studies. However, there are particular study stages available for modelling one-way coupled fluid–structure interaction. The fluid–structure interaction interface applicable to 3D geometry models adds the following default nodes to the Model Builder: (a) wall (for the fluid), (b) prescribed mesh displacement (for the mesh movement), (c) free (for the solid mechanics, which initially is not applicable to any boundary because the default settings assume a fluid domain), and (d) initial values. Furthermore, a fluid–shell interface boundary node adds the fluid–structure interaction to the fluid–shell boundary. Only inner fluid–shell boundaries can use this node. A saddle-point feature exists in the linearised Navier–Stokes equation system unless the density is pressure-dependent. This indicates that there are zeros on the diagonal of the Jacobian matrix. The equation system essentially shares many numerical aspects with a saddle-point system, even in cases where the density depends on the pressure. The default solution recommendation for tiny 2D and 3D models is a direct solver. For small models, direct solutions are incredibly quick and robust, and they can handle the majority of nonsingular systems. Regretfully, when the model grows larger, they become slower, and their memory needs range from N1.5 to N2, where N is the total number of degrees of freedom in the model. For large 2D and 3D models, the iterative GMRES solver is, therefore, the default recommendation. Smoothers are necessary for multigrid approaches, but the number of suitable smoothers is limited by the linear system’s saddle-point nature. Numerical stabilisation has a major impact on the smoother’s efficiency. When diffusion is involved, iterative solutions perform at their best. SCGS is the standard smoother for P1 + P1 components. This smoother is strong and effective and specifically made to solve saddle-point systems on models with anisotropic elements. Even in the absence of crosswind dispersion, the SCGS smoother performs admirably. Higher-order elements may occasionally operate with SCGS, particularly if the method is set to mesh element lines in the SCGS options. By definition, a stable formulation has no time derivatives.

$$\rho (\mathrm{u}\cdot \nabla )\mathrm{u}=\nabla \cdot \left[-p\mathrm{I}+\mu \left(\nabla \mathrm{u}+(\nabla \mathrm{u}{)}^T\right)\right]+\mathrm{F}$$

(19)

It is necessary to make an initial guess that is reasonably close to the solution in order to solve the above equation. If such an estimate is not available, a transitory solution can be used in its place. However, in terms of computational time, this is a somewhat expensive strategy. Adding a fake time derivative to (19) is an intermediate method:

$$\rho {\displaystyle \frac{\mathrm{u}-\mathrm{n}\mathrm{o}\mathrm{j}\mathrm{a}\mathrm{c}\left(\mathrm{u}\right)}{\mathsf{\Delta }\tilde{t}}}+\rho \left(\mathrm{u}\cdot \nabla \right)\mathrm{u}=\nabla \cdot \left[-p\mathrm{I}+\mu \left(\nabla \mathrm{u}+(\nabla \mathrm{u}{)}^T\right)\right]+\mathrm{F}$$

(20)

A pseudo-time step is located. The final answer is unaffected by this term because u − nojac (u) is always zero. However, it does have an impact on the system of discrete equations and converts a nonlinear iteration into a step of a time-dependent solver. By default, pseudo-time stepping is not enabled. Each element’s pseudo-time step can be selected separately depending on the local CFL number.

$$\mathsf{\Delta }\tilde{t}={\mathrm{C}\mathrm{F}\mathrm{L}}_{\mathrm{l}\mathrm{o}\mathrm{c}}{\displaystyle \frac{h}{\left|\mathrm{u}\right|}}$$

(21)

The mesh cell size is denoted by h. A small time step corresponds to a small CFL number. Starting small and increasing the CFL number progressively as the solution approaches a steady state is a viable technique. The automated setting recommends a PID regulator for the pseudo-time step in the default solver if the automatic expression for CFLloc is set to the built-in variable CFLCMP. As the solution approaches convergence, the PID regulator raises CFLloc from its initial low value.

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The default manual expression is 1.3min niterCMP ( ), 9 + if niterCMP > 20. 9 1.3 min niterCMP ( ) − 20. 9 ( ) 0 + if niterCMP > 40. 90 1.3 min niter CMP ( ) − 40, 9 ( );

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The nonlinear iteration number is stored in the variable niterCMP. For the first nonlinear iteration, it is equal to one. CFLloc is initially at 1.3 and rises by 30% every iteration, reaching 1.39 ≈ 10.6. It stays there until iteration number 20, at which point it begins to rise and eventually reaches about 10.6. Following iteration number 40, a final rise brings it to 10.6. For some sophisticated flows, Equations (3) to (19) may cause CFLloc to increase too quickly or too slowly. After that, CFLloc can be adjusted for a particular use.

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The pressure field, fluid–slosh velocity field, and water volume distribution of the VOF model inside the disco-rectangular model were obtained every 0.005 s for the three different materials—nylon, polyamide, and polylactic acid (PLA)—of two overhead baffles fixed at regular intervals over half of the disco-rectangular tank. These results were obtained following a dry run of the numerical simulation carrying a time step of range of (0.00005).

5. Results

Sloshing time t = 0.1 s Pressure distribution (Figure 8).

Sloshing time t = 0.1 s Velocity distribution (Figure 9).

6. Discussion

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The pressure contours of the VOF model results are shown in Figure 8. The results taken for a sloshing time of t = 0 to 1 s are for the three different material baffles made from nylon, polyamide, and PLA (polylactic acid). From the pressure contour plots, it can be viewed that nylon baffles deflect more compared to polyamide and PLA baffles at a time of t = 0.1 to 0.5 s. The pressure of the fluid sloshing increases to a peak in the initial stages, and after t = 0.5 s to 1 s, the pressure of the fluid decreases for nylon but increases for the polyamide and PLA baffles:

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Figure 9 illustrates the velocity contours of liquid-sloshing action in the disco-rectangular tank for the nylon, polyamide, and PLA-type baffles. From this result, the sloshing speed reduces for the nylon baffles from a time of 0.5 to 1 s, whereas it increases for the polyamide baffles and shows moderate variations for the PLA baffles.

The nylon overhead baffle VOF model results were considered. Figure 10a represents the water volume distribution inside the tank, where the volume of fluid increases suddenly at a sloshing time of t = 0.1 s and then comes to rest at t = 1 s. This is because of the nylon linear elastic material, which resists a larger load with maximum displacement initially, and the sloshing speed decreases further as the water volume of the fluid tends to slow down by the second and third oscillation of the overhead baffles, which do not strike each other. The fluid depth in the initial conditions is high, but the fluid depth in the final conditions after sloshing is half. Figure 10b represents the directional displacement of nylon baffles in the x, y, and z directions that yield the maximum deflections from a sloshing time of t = 0.01 s to t = 0.2 s. As the overhead baffles made of nylon yield to the linear elasticity, they respond to the fluid movement, and the fluid comes to rest with a decrease in the frequency of the displaced baffles from a sloshing time of t = 0.5 s to t = 1 s as it acts as a damper.

A similar size of tank and overhead baffles were used in a study of different materials to investigate polyamide and PLA baffle behaviour in a VOF model in a disco-rectangular tank. Figure 11 describes the water volume distribution and directional displacement of polyamide baffles. With these polyamide baffles, the water volume distribution inside the tank never stops and sloshing time exceeds 1 s, with a high natural frequency exhibited by the polyamide baffles.

Similarly, Figure 12 describes the water volume distribution and directional displacement of PLA overhead baffles. The results of water volume distribution from a sloshing time of t = 0.1 s to t = 0.5 s slowly increase as the PLA material exhibits normal elastic behaviour compared to nylon and polyamide baffles, where the oscillation increases slowly due to the increasing displacement. This increase in displacement will stop only when the fluid sloshing comes to rest at t = $\infty$. Thus, the natural frequency of the PLA baffles is very high compared to the polyamide baffles and provides overdamped oscillation.

Figure 13 finds that nylon baffles show a larger volume of displacement magnitude, whereas polyamide and PLA show a smaller volume of displacement magnitude. This effect is due to their material elasticity, plasticity of polymers, and arrangement of chain links. Nylon baffles possess good elasticity and moderate plasticity that can yield a maximum displacement of 3 mm, whereas polyamide baffles possess moderate elasticity and true plasticity and yield a maximum displacement of 0.045 mm. However, PLA baffles possess low elasticity and high plasticity, which, in turn, yield a maximum displacement of 0.14 mm. Thus, the PLA baffles displace 4.6% of nylon with effective plasticity, and their plasticity behaviour causes them to oscillate in overdamped cases, and the sloshing of the tank never stops. An intermediate force is needed to stop PLA baffles in overdamped or underdamped conditions.

The results of the natural frequency of excited baffles under sloshing were recorded for a sloshing time of t = 0 to 1 s and are plotted in Figure 14. The natural frequency initially is increases for the nylon, polyamide, and PLA-type double baffles from 0 to 0.5 s. After that, the natural frequency decreases. This represents the ideology of material that possesses good elasticity compared to plasticity and good damping assistance for sloshing liquid under the consideration of the roughness ratio of the material for fluid with a viscous effect. The elastic material nylon possesses a low natural frequency response, as it shows higher displacement compared to other baffle types, and the sloshing effect is reduced. The material polyamide possesses a high natural frequency response as it shows lower displacement, and the sloshing effect continues. The fluid does not come to rest; hence, it is overdamped. The material PLA possesses a moderate natural frequency response as it shows moderate displacement, and the sloshing effect is reduced lower than nylon baffles.

The fluid–structure interaction of the simplified VOF model for the nylon–nylon baffles in the disco-rectangular tank is shown in Figure 15. The nylon–nylon baffles show the maximum displacement with respect to a sloshing time of 0.1 s and yield the minimum displacement at a sloshing time of 0.5 s. Neutralisation occurs at a sloshing time of 0.6 s. Similarly, the fluid–structure interaction of the simplified VOF model for the PLA-PLA and polyamide–polyamide baffles was studied.

7. Conclusions

✓

From several numerical studies on sloshing, we conclude that nylon baffles show a good response to sloshing arrest for about 1 s, whereas the polyamide and PLA baffles show a poor response to sloshing as it exceeds 1 s.

✓

The material that possesses good elasticity with moderate plasticity yields a low natural frequency response to sloshing emerging from a liquid sliding-door open system. This is owing to the maximum deflection possessed by the elastic material;

✓

Similarly, nylon baffles yield the maximum deflection with a low-frequency response for the sliding door opening action system for liquid sloshing;

✓

VOF modelling of the different pairs of baffles configured in a disco-rectangular tank, integrating their displacement effect and interaction with fluid, were studied;

✓

Among all three pairs of baffles, the nylon–nylon baffle configuration yields the best plasticity and poor surface roughness to respond to the sloshing phenomenon and be in a state of control and damper.

However, the material that possesses good plasticity compared to elasticity considered here does not show a good response to sloshing arrest motion. Here, we investigated good plasticity materials, like polyamide and PLA, for arresting the sloshing liquid to rest. However, polyamide and PLA did not meet the requirements as they exhibit high natural frequency excitation, and the sloshing time continues beyond the resting action. Therefore, the material that possesses good plasticity with low surface roughness is meant for viscous fluid-sloshing control in the long run. The sloshing time increases for this type of material irrespective of its surface roughness and viscous fluid.

Investigations like changing the type of pair of baffles with two different materials and liquid sloshing times are to be carried out in future to find out if a combined double-baffle response to sloshing behaviour is needed.