Damping of liquid sloshing in the tanks subjected to vertical acceleration by using the boundary element method
Abstract
The paper presents numerical simulations of liquid sloshing in the partially filled fuel tanks subjected to vertical acceleration. The tanks are considered to be shells of revolution, and a liquid inside the tank is supposed to be incompressible with viscosity effects being accounted for. The liquid motion is irrotational, and a velocity potential can be introduced. The boundary value problem is formulated for the Laplace’s equation to obtain the velocity potential and the free surface level. Non-penetration boundary conditions are used at the wetted surface of a shell. The kinematic and dynamic boundary conditions are given on the free liquid surface. Effects of a surface tension are included into the Bernoulli equation as an additional pressure that is proportional to the mean curvature of the free surface. It allows considering coupled effects of both gravitational and capillarity waves. The boundary value problem is solved by using boundary element method. The system of the Mathieu equations is obtained and modified according to the damping effects. These effects are estimated, and stability regions on Ince-Strutt diagram are specified.
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References
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Karaiev, A., et al. Axisymmetric polyharmonic spline approximation in the dual reciprocity method. ZAMM‐Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik , Vol. 101, No.1, e201800339, DOI:10.1002/zamm.201800339
Dodge F. T. The New «Dynamic Behavior of Liquids in Moving Containers», San Antonio, Texas : Southwest Research Inst., 2000.
M. Ramírez Barrios, J. Collado, and F. Dohnal, "Stability of Coupled and Damped Mathieu Equations Utilizing Symplectic Properties," in Nonlinear Dynamics of Structures, Systems and Devices, Cham: Springer International Publishing, 2020, pp. 137–145. Accessed: Oct. 16, 2021. [Online]. Available: https://doi.org/10.1007/978-3-030-34713-0_14
Housner G. W. Dynamic pressures on accelerated fluid containers. Bulletin of the Seismological Society of America. 1957. Vol. 47, no. 1. P. 15–35. URL: https://doi.org/10.1785/bssa0470010015 (дата звернення: 16.10.2021).
Housner G. W. The dynamic behavior of water tanks. Bulletin of the Seismological Society of America. 1963. Vol. 53, no. 2. P. 381–387. URL: https://doi.org/10.1785/bssa0530020381 (дата звернення: 16.10.2021).
The dynamic behavior of liquids in moving containers: With applications to space vehicle technology. Washington : Scientific and Technical Information Division, National Aeronautics and Space Administration; [for sale by Supt. of Docs., U. S. Govt. Print. Off.], 1966. 467 p.
Ibrahim R. A. Liquid sloshing dynamics: Theory and applications. New York : Cambridge University Press, 2005.
Kolmanovskii V., Myshkis A. Applied Theory of Functional Differential Equations. Dordrecht : Springer Netherlands, 1992. URL: https://doi.org/10.1007/978-94-015-8084-7 (дата звернення: 16.10.2021).
Myshkis A. D. Low-Gravity Fluid Mechanics: Mathematical Theory of Capillary Phenomena. Springer, 2011. 604 p.
Microgravity validation of a novel system for RNA isolation and multiplex quantitative real time PCR analysis of gene expression on the International Space Station / M. Parra et al. PLOS ONE. 2017. Vol. 12, no. 9. P. e0183480. URL: https://doi.org/10.1371/journal.pone.0183480 (дата звернення: 16.10.2021).
BEM analysis of gravitational–capillarity waves on free surfaces of compound shells of revolution / V. I. Gnitko et al. International Journal of Computational Methods and Experimental Measurements. 2021. Vol. 9, no. 1. P. 38–50. URL: https://doi.org/10.2495/cmem-v9-n1-38-50 (дата звернення: 16.10.2021).
Gnitko V., et al., Multi-Domain Boundary Element Method for Axisymmetric Problems in Potential Theory and Linear Isotropic Elasticity, BEM/MRM 2018, WIT Press, 2018, pp.13-25, DOI:10.2495/be410021.
The effect of liquid viscosity on sloshing characteristics / C.-F. Zou et al. Journal of Marine Science and Technology. 2015. Vol. 20, no. 4. P. 765–775. URL: https://doi.org/10.1007/s00773-015-0329-y (дата звернення: 16.10.2021).
High Frequency Sloshing - Energy Dissipation and Viscous Damping through CFD / L. F. Blas Martinez et al. WCX™ 17: SAE World Congress Experience. 400 Commonwealth Drive, Warrendale, PA, United States, 2017. URL: https://doi.org/10.4271/2017-01-1317 (дата звернення: 16.10.2021).
Demirel E., Aral M. Liquid Sloshing Damping in an Accelerated Tank Using a Novel Slot-Baffle Design. Water. 2018. Vol. 10, no. 11. P. 1565. URL: https://doi.org/10.3390/w10111565 (дата звернення: 16.10.2021).
Dennehy C. Recent Experiences of the NASA Engineering & Safety Center (NESC) GN&C Technical Discipline Team (TDT). AIAA Guidance, Navigation, and Control Conference, Toronto, Ontario, Canada. Reston, Virigina, 2010. URL: https://doi.org/10.2514/6.2010-8427 (дата звернення: 16.10.2021).
Boundary element method in nonlinear sloshing analysis for shells of revolution under longitudinal excitations / E. Strelnikova et al. Engineering Analysis with Boundary Elements. 2020. Vol. 111. P. 78–87. URL: https://doi.org/10.1016/j.enganabound.2019.10.008 (дата звернення: 16.10.2021).
BEM analysis of gravitational–capillarity waves on free surfaces of compound shells of revolution / V. I. Gnitko et al. International Journal of Computational Methods and Experimental Measurements. 2021. Vol. 9, no. 1. P. 38–50. URL: https://doi.org/10.2495/cmem-v9-n1-38-50 (дата звернення: 16.10.2021).
Strelnikova, E., et al. Free and Forced Vibrations of Liquid Storage Tanks with Baffles J. Modern Technology & Engineering, Vol.3, No.1, 2018, pp.15-52. URL: http://jomardpublishing.com/UploadFiles/Files/journals/JTME/V3No1/StrelnikovaE.pdf (дата звернення: 16.10.2021).
Naumenko, V., et al. Singular integrals accuracy of calculations in two-dimensional problems using boundary element methods. Engineering analysis with boundary elements, Vol. 26, No.1, 2002, pp.95-98. DOI:10.1016/S0955-7997(01)00041-8
Karaiev, A., et al. Axisymmetric polyharmonic spline approximation in the dual reciprocity method. ZAMM‐Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik , Vol. 101, No.1, e201800339, DOI:10.1002/zamm.201800339
Dodge F. T. The New «Dynamic Behavior of Liquids in Moving Containers», San Antonio, Texas : Southwest Research Inst., 2000.
Ramírez Barrios M., Collado J., Dohnal F. Stability of Coupled and Damped Mathieu Equations Utilizing Symplectic Properties. Nonlinear Dynamics of Structures, Systems and Devices. Cham, 2020. P. 137–145. URL: https://doi.org/10.1007/978-3-030-34713-0_14 (дата звернення: 16.10.2021).
