Analysis of Damping of Fluid Oscillations in Spherical Tanks Using the Boundary Element Method

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DOI https://doi.org/10.15407/pmach2026.01.035
Journal Journal of Mechanical Engineering – Problemy Mashynobuduvannia
Publisher Anatolii Pidhornyi Institute of Power Machines and Systems
of National Academy of Science of Ukraine
ISSN  2709-2984 (Print), 2709-2992 (Online)
Issue Vol. 29, no. 1, 2026 (March)
Pages 35-46
Cited by J. of Mech. Eng., 2026, vol. 29, no. 1, pp. 35-46

 

Authors

Vasyl I. Hnitko, Anatolii Pidhornyi Institute of Power Machines and Systems of NAS of Ukraine (2/10, Komunalnykiv str., Kharkiv, 61046, Ukraine), e-mail: gnitkovi@gmail.com, ORCID: 0000-0003-2475-5486

Kyrylo H. Dehtiarov, Anatolii Pidhornyi Institute of Power Machines and Systems of NAS of Ukraine (2/10, Komunalnykiv str., Kharkiv, 61046, Ukraine), e-mail: kdegt89@gmail.com, ORCID: 0000-0002-4486-2468

Andrii S. Kolodiazhnyi, Anatolii Pidhornyi Institute of Power Machines and Systems of NAS of Ukraine (2/10, Komunalnykiv str., Kharkiv, 61046, Ukraine), e-mail: 7ask7@ukr.net, ORCID: 0009-0008-4026-6715

Denys V. Kriutchenko, Anatolii Pidhornyi Institute of Power Machines and Systems of NAS of Ukraine (2/10, Komunalnykiv str., Kharkiv, 61046, Ukraine), e-mail: wollydenis@gmail.com, ORCID: 0000-0002-6804-6991

 

Abstract

The aim of this study is to develop numerical methods for analyzing the stability of fluid motion in spherical tanks with horizontal baffles. Partially filled spherical tanks are important components of modern engineering systems. They are widely used as storage vessels for drinking water and hazardous liquids, as well as structural elements of launch vehicle fuel tanks. Experimental testing of such tanks for strength and dynamic stability is generally expensive and not always safe. This necessitates the development of virtual testing methods based on efficient computational algorithms. In this context, the development of new numerical methods for analyzing fluid oscillations and motion stability in tanks, where the radius of the free surface depends on the filling level, is a relevant problem. The study employs methods of potential theory, the boundary element method, the method of prescribed normal modes, and numerical techniques for solving systems of differential equations. Spectral boundary value problems are solved to determine the natural frequencies and mode shapes of fluid oscillations in spherical tanks without baffles and in tanks with horizontal baffles containing openings of various diameters. These problems are reduced to systems of one-dimensional singular integral equations. The obtained natural modes are used as basis functions for solving the problem of forced fluid oscillations in spherical tanks subjected to simultaneous vertical and horizontal excitations. Expressions for the velocity potential and the free surface elevation function are derived in the form of infinite series, and the convergence of these series is analyzed. The problem of determining the dynamic characteristics of the fluid is reduced to solving a system of ordinary differential equations of the Mathieu type, which makes it possible to study the stability of fluid motion in a spherical tank under combined horizontal and vertical loading. An efficient numerical approach for studying fluid oscillations and motion stability in partially filled spherical tanks has been developed and implemented. The proposed approach can be used for virtual testing of spherical tanks and for analyzing fluid behavior in the design and operation of tanks and fuel systems in aerospace engineering.

 

Keywords: fluid oscillations, damping, spherical tanks, horizontal baffles, singular integral equations, boundary element method, stability of fluid motion.

 

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References

  1. Housner, G. W. (1957). Dynamic pressures on accelerated fluid containers. Bulletin of the Seismological Society of America, vol. 47, iss. 1, pp. 15–35. https://doi.org/10.1785/BSSA0470010015.
  2. Housner, G. W. (1963). Dynamic behavior of water tanks. Bulletin of Seismological Society of America, vol. 53, iss. 2, pp. 381–387. https://doi.org/10.1785/BSSA0530020381.
  3. Ibrahim, R. A. (2005). Liquid sloshing dynamics. Theory and applications. Cambridge University Press, 972 p. https://doi.org/10.1017/CBO9780511536656.
  4. Veletsos, A. S. & Yang, J. Y. (1976). Dynamics of fixed-base liquid-storage tanks. In: Proceedings of U.S.–Japan Seminar on Earthquake Engineering Research with Emphasis on Lifeline Systems, (Tokyo, Japan, November, 1976), vol. 8–1, pp. 317–341.
  5. Sun, S. Y. & Wu, G. X. (2024). Sloshing in a tank with elastic side walls and a membrane cover. Physics of Fluids, vol. 36, iss. 10, article 102127. https://doi.org/10.1063/5.0238210.
  6. Shvets, A., Murawski, K., & Fedorov, Y. (2025). Analytical determination of critical forces during buckling of systems consisting of two pinned connected rods. Meccanica, vol. 60, pp. 441–455. https://doi.org/10.1007/s11012-025-01941-3.
  7. Gnitko, V. I., Karaiev, A. O., Degtyariov, K. G., Vierushkin, I. A., & Strelnikova, E. A. (2022). Singular and hypersingular integral equations in fluid–structure interaction analysis. WIT Transactions on Engineering Sciences, vol. 134, pp. 67–79. https://doi.org/10.2495/BE450061.
  8. Smetankina, N. & Pavlikov, V. (2021). Mathematical model of the stress state of the antenna radome joint with the load-bearing edging of the skin cutout. In: Cioboată D. D. (eds) International Conference on Reliable Systems Engineering (ICoRSE) – 2021. ICoRSE 2021. Lecture Notes in Networks and Systems, vol. 305, pp. 287–295. https://doi.org/10.1007/978-3-030-83368-8_28.
  9. Smetankina, N., Merkulova, A., Merkulov, D., Misiura, S.,& Misiura, I. (2020). Modelling thermal stresses in laminated aircraft elements of a complex form with account of heat sources. In: Cioboată, D. D. (eds) International Conference on Reliable Systems Engineering (ICoRSE) – 2022. ICoRSE 2022. Lecture Notes in Networks and Systems, vol. 534, pp. 233–246. https://doi.org/10.1007/978-3-031-15944-2_22.
  10. Strelnikova, E., Kriutchenko, D., Gnitko, V., & Tonkonozhenko, A. (2020). Liquid vibrations in cylindrical tanks with and without baffles under lateral and longitudinal excitations. International Journal of Applied Mechanics and Engineering, vol. 25, iss. 3, pp. 117–132. https://doi.org/10.2478/ijame-2020-0038.
  11. Strelnikova, E., Kriutchenko, D., & Gnitko, V. (2019). Liquid vibrations in cylindrical quarter tank subjected to harmonic, impulse and seismic lateral excitations. Journal of Mathematics and Statistical Science, vol. 5, iss. , pp. 31–41.
  12. Choudhary, N., Bora, S. N., & Strelnikova, E. (2021). Study on liquid sloshing in an annular rigid circular cylindrical tank with damping device placed in liquid domain. Journal of Vibration Engineering & Technologies, vol. 9, pp. 1577–1589. https://doi.org/10.1007/s42417-021-00314-w.
  13. Balas, O.-M., Doicin, C. V., & Cipu, E. C. (2023). Analytical and numerical model of sloshing in a rectangular tank subjected to a braking. Mathematics, vol. 11, iss. 4, pp. 949–955. https://doi.org/10.3390/math11040949.
  14. Liu, L. & Li, J. (2022). Dynamic deformation and perforation of ellipsoidal thin shell impacted by flat-nose projectile. Materials, vol. 15, iss. 12, article 4124. https://doi.org/10.3390/ma15124124.
  15. Sierikova, O., Strelnikova, E., & Degtyarev, K. (2022). Seismic loads influence treatment on the liquid hydrocarbon storage tanks made of nanocomposite materials. WSEAS Transactions on Applied and Theoretical Mechanics, vol. 17, pp. 62–70. https://doi.org/10.37394/232011.2022.17.9.
  16. Martinez-Carrascal, J. & González-Gutiérrez, L. (2021). Experimental study of the liquid damping effects on a SDOF vertical sloshing tank. Journal of Fluids and Structures, vol. 100, article 103172. https://doi.org/10.1016/j.jfluidstructs.2020.103172.
  17. Gani, E., Öztürk, S., & Sari, A. (2025). Effects of liquid sloshing in storage tanks. An overview of analytical, numerical, and experimental studies. International Journal of Steel Structures, vol. 25, pp. 544–556. https://doi.org/10.1007/s13296-025-00946-8.
  18. Lu, T. & Cao, D. (2025). Comparative study on wave response to vertical baffle orientation for resonant sloshing suppression in an upright cylindrical tank. Ocean Engineering, vol. 341, part 2, article 122526. https://doi.org/10.1016/j.oceaneng.2025.122526.
  19. Degtyariov, K., Kriutchenko, D., Osypov, I., Sierikova, O., & Strelnikova, E. (2024). Dampers influence on sloshing mitigation in fuel tanks of launch vehicles and reservoirs. In: Nechyporuk, M., Pavlikov, V., Krytskyi, D. (eds) Integrated Computer Technologies in Mechanical Engineering – 2023. ICTM 2023. Lecture Notes in Networks and Systems, vol. 1008, pp. 420–430. https://doi.org/10.1007/978-3-031-61415-6_36.
  20. Raynovskyy, I. A. & Timokha, A. N. (2020). Sloshing in upright circular containers: Theory, analytical solutions, and applications. CRC Press, 170 p. https://doi.org/10.1201/9780429356711.
  21. Brebbia, C. A., Telles, J. C. F, & Wrobel, L. C. (1984). Boundary element techniques. Berlin and New York: Springer-Verlag. https://doi.org/10.1007/978-3-642-48860-3
  22. Brebbia, C. A. & Domínguez, J. (1992). Boundary elements: An introductory course. Computational Mechanics Publications / McGraw-Hill.
  23. Gradshteyn, I. S. & Ryzhik, I. M. (2007). Table of integrals, series, and products. Amsterdam: Academic Press.
  24. Sierikova, O., Strelnikova, E., & Degtyariov, K. (2022). Strength characteristics of liquid storage tanks with nanocomposites as reservoir materials. 2022 IEEE 3rd KhPI Week on Advanced Technology (KhPIWeek), Kharkiv, Ukraine, pp. 1–7. https://doi.org/10.1109/KhPIWeek57572.2022.9916369.
  25. Kovacic, I., Rand, R. H., & Sah, S. M. (2018). Mathieu’s equation and its generalizations: Overview of stability charts and their features. Applied Mechanics Reviews, vol. 70, iss. 2, article 020802. https://doi.org/10.1115/1.4039144.
  26. Mciver, P. (1989). Sloshing frequencies for cylindrical and spherical containers filled to an arbitrary depth. Journal of Fluid Mechanics, vol. 201, pp. 243–257. https://doi.org/10.1017/S0022112089000923.
  27. Yang, H., Purandare, Q., Peugeot, R., & West, J. (2012). Prediction of liquid slosh damping using a high resolution CFD tool. 48th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, USA, Georgia, Atlanta, 30 July–01 August 2012, pp. 1–14. https://doi.org/10.2514/6.2012-4294.

 

Received 02 February 2026
Accepted 12 March 2026
Published 30 March 2026