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Sloshing reduction in a swaying tank with porous baffles using scaled boundary finite element method
Pramod, Aln, Choudhury, Aditi, Vijay, KG, Natarajan, S
TL;DR
The paper addresses sloshing-induced loads in partially filled tanks and the limited use of SBFEM for porous-baffle configurations. It extends SBFEM within linear potential flow theory, models baffle flow with Darcy’s law, and finds accurate, computationally efficient predictions with top-mounted convex baffles most effective overall.
Problem
Sloshing in partially filled tanks can produce adverse loading, while investigations of sloshing dynamics using SBFEM remain scanty.
Method
The paper extends SBFEM to rectangular swaying tanks with multiple porous baffles using linear potential flow theory, Darcy’s law, minimally divided subdomains, and higher-order edge polynomials.
Results
Top-mounted convex baffles provide the most effective overall sloshing suppression across modes, while the framework agrees well with literature results and uses at least one order fewer degrees of freedom than traditional approaches.
Takeaways & Limitations
The proposed SBFEM framework offers an accurate and computationally reduced approach for evaluating porous-baffle configurations and their effects on sloshing response.
Takeaways & Limitations
The formulation assumes linear potential flow and Darcy’s law for flow through thin, rigid porous baffles.
Abstract
from arXiv · showhide
Sloshing is an inevitable phenomenon in an ocean-going vessel that can have adverse effects. In this work, the mitigation of sloshing is investigated using multiple thin porous baffles of various configurations in a partially filled swaying tank. The boundary value problem is solved within the framework of a linearized potential flow theory using the scaled boundary finite element method (SBFEM). The flow through the thin porous baffles is assumed to follow Darcy's law. The computational domain is divided into a minimum number of subdomains due to the presence of porous baffles and to ensure star convexity. Higher-order polynomials are used along each subdomain edge to represent the unknown field, i.e., velocity potential. The developed numerical model is validated with the known results in the literature. Subsequently, various results, such as the amplification factor and the forces of the tank wall, are presented and discussed for the effect of configuration, porosity, slosh tank width, depth of baffle submergence and the space between adjacent baffles. From the parametric study, it is observed that top-mounted baffles enhance sloshing suppression by $50\%$ compared to bottom-mounted vertical baffles, considering all sloshing modes. Assessing the overall effectiveness, top-mounted convex baffle configuration emerges as the most efficient configuration for sloshing suppression, achieving a well-balanced reduction across all modes.
1. Introduction
The paper frames liquid sloshing as a damaging free-surface motion and motivates porous baffles and SBFEM as mitigation and modeling approaches. It addresses limited prior SBFEM investigations by modeling tanks with multiple porous-baffle configurations.
- Motivation: Sloshing is an externally induced free-surface motion in partially filled containers that can load tank walls and cause structural damage.The introduction identifies ocean-going vessels and other systems as affected applications.
- Mitigation: Porous baffles are preferred to impermeable baffles because they enhance damping and suppress dynamic tank forces.Their configuration, porosity, length, and tank dimensions can tune free-surface elevation, frequency, and amplitude.
- Prior approaches: Existing analytical solutions are limited to simple geometries and baffle configurations, motivating finite-element and boundary-based approaches.Prior work applied FEM and BEM to broaden the geometries and formulations available for sloshing analysis.
- SBFEM: SBFEM solves the circumferential direction numerically and the radial direction analytically, while supporting polygonal elements and polynomial approximations.The method has also been applied to several other dynamics and mechanics problems.
- Contribution: The paper proposes SBFEM for sloshing in tanks with multiple porous baffles, using Darcy’s law and linear potential flow theory.The domain is divided into subdomains around the baffles, with higher-order polynomials along subdomain edges; the authors report lower computational effort without compromising accuracy.
2. Mathematical Formulation
The formulation models two-dimensional sloshing in a rectangular tank with multiple porous baffles using linear potential flow, Darcy’s law, and configuration-specific parabolic geometries. It defines the governing boundary conditions, porosity limits, response measures, and baffle arrangements used in the study.
- Tank and baffle geometry: The tank is modeled as a two-dimensional rectangle of dimensions 2a×h with multiple vertical porous baffles whose endpoints follow parabolic profiles.Convex and concave profiles determine whether the center or extreme-side baffles are shorter or taller.
- Governing formulation: The velocity potential is assumed harmonic, with angular frequency ω describing the sloshing response.The potential is represented through a spatial field multiplied by a time-harmonic factor.
- Boundary conditions: The governing boundary conditions impose zero normal flux at the tank bottom and unit normal derivative at the swaying tank walls.These conditions accompany the governing equations for two-dimensional sloshing.
- Porous-baffle condition: Darcy’s law makes flow across each thin porous baffle linearly proportional to the pressure difference between its two sides.The baffle relation uses the velocity potentials ϕ+ and ϕ− on either side and introduces a porous-effect parameter σ.
- Wave relation: The wave number k1 is obtained from the dispersion relation linking k1, the fill depth h, angular frequency ω, and gravity g.The dispersion relation determines the wave-number quantity used in the sloshing formulation.
- Porosity: The porosity parameter spans impermeable baffles at b = 0 to infinitely permeable baffles at b →∞, equivalent to no baffle.This parameterization supports comparison across different baffle permeabilities.
- Response measures: The formulation evaluates wave elevation, normalized amplification factor, wall pressure, and integrated sloshing force as response quantities.These quantities are used to characterize the tank response under the modeled configurations.
- Baffle configurations: Four cases combine top- or bottom-mounted baffles with concave or convex parabolic profiles.The cases differ in the parabola equation and the normalized middle, intermediate, and extreme-side baffle heights.
3. Overview of the scaled boundary finite element method
The SBFEM converts the sloshing boundary-value problem into a semi-analytical formulation by treating the radial direction analytically and the circumferential boundary numerically. Boundary geometry and field variables are interpolated with shape functions, and element-level coefficient matrices are assembled to obtain the governing differential system.
- Scaled-boundary coordinates: SBFEM defines a scaling centre from which the computational domain is directly visible and introduces radial-circumferential coordinates.The method reduces the governing partial differential equations to ordinary differential equations using a variational framework.
- Semi-analytical treatment: The radial solution is represented analytically, while numerical interpolation is applied around the circumferential boundary.This separates the analytical and numerical treatments of the domain.
- Boundary interpolation: The boundary geometry is interpolated with shape functions using boundary coordinates and nodal points.The mapping uses the radial coordinate ξ, boundary coordinates xI, and the shape-function matrix N(η).
- Field approximation: The potential field is approximated with the same shape functions used to represent the polygonal geometry.This provides the field interpolation required for the variational formulation.
- Element formulation: Variational formulation produces a differential equation for the nodal pressure amplitude using scaled-boundary coefficient matrices assembled element-by-element.The matrices E0, E1, and E2 are evaluated over element or subdomain boundaries.
- System solution: The resulting second-order system is converted to a first-order system with twice as many unknowns and a Hamiltonian matrix.The conversion introduces qh(ξ) and leads to the corresponding stiffness formulation.
0 ET 1 E1E−1
The SBFEM formulation obtains the unknown potential field through eigenvalue decomposition of the Hamiltonian matrix, retaining bounded modes for polygonal subdomains. Nodal potentials determine integration constants and enable computation of nodal forces and the polygon stiffness matrix.
- The unknown potential field is obtained by eigenvalue decomposition of the Hamiltonian matrix.
- The eigenvectors represent potential and potential-gradient components, while the eigenvalues are partitioned into negative and positive sets.
- For bounded polygons, only the first n negative eigenvalues and corresponding eigenvectors are retained to ensure a finite, bounded solution at the scaling center.
- Integration constants are determined from the nodal potential solution, with ϕb = ϕh(ξ = 1).
- The nodal force vector is computed by evaluating the boundary variable at ξ = 1 after substituting the potential-field expression.
- The polygon stiffness matrix is computed from the Hamiltonian matrix eigenvectors.
4. Numerical examples
The numerical examples assess convergence, validation, baffle configuration, porosity, tank width, submergence depth, and baffle spacing using amplification and wall-force responses. Top-mounted configurations and selected geometric and porosity parameters provide stronger or more balanced sloshing suppression across modes.
- Convergence and validation: The computational domain is minimally divided into star-convex subdomains, with boundary shape-function order varied to assess accuracy and convergence.The study uses higher-order boundary polynomials within the SBFEM framework.
- Convergence and validation: The proposed framework shows very good agreement with boundary-element results for normalized amplification and sloshing force, using a matrix size of 94.Results are identical at x = −a and x = a, consistent with the cited reference.
- Baffle configuration: Case-C provides the most effective overall sloshing reduction among the tested configurations by balancing performance across the first, third, and fifth modes.Bottom-mounted baffles attenuate surface waves less effectively because they primarily affect flow near the tank base.
- Porosity: Porosity values 0.1 and 0.2 provide the best performance for reducing sloshing amplitude, while lower porosity notably diminishes the fifth-mode amplification peak.The reported trend is attributed to enhanced wave-energy dissipation through redistributed flow interactions.
- Tank width: Increasing tank width produces more resonance peaks at lower frequencies and increases free-surface displacement, amplification factor, and wall loading.The comparison fixes S/a = 0.3, d/h = 0.8, d2/h = 0.4, and P = 0.2 while varying a/h = 6, 4, and 2.
- Submergence depth: A middle-baffle height of d = 0.8h gives more effective suppression across multiple modes than the lower tested heights.Increasing baffle height reduces the first-, third-, and fifth-mode resonance peak amplitudes, although the largest peak shifts to the third mode at d = 0.8h.
- Baffle spacing: Spacing S = 0.4a yields the lowest third-resonance sloshing amplitude and force, whereas the first two resonance peaks vary minimally with spacing.The comparison uses a/h = 4.0, P = 0.2, d/h = 0.8, and d2/h = 0.4.
5. Conclusions
The extended SBFEM framework models sloshing in rectangular tanks with porous baffles while reducing computational cost. Parametric analyses identify effective baffle designs and operating parameters for sloshing reduction.
- The framework discretizes only domain boundaries, requiring at least one order fewer degrees of freedom than traditional approaches without compromising accuracy.A fifth-order polynomial provides accurate results.
- Different baffle arrangements are analyzed to identify more effective configurations for sloshing suppression.
- Optimal porosity, tank width-to-height ratio, submergence depth, and baffle spacing are determined for maximum sloshing reduction.
- The numerical results are planned for further validation through experimental investigations.