Mechanical model of low-viscosity liquid sloshing with capillary effects

DOI: 10.34759/trd-2023-129-12

Аuthors

Temnov A. N.^*, Shkapov P. M.^**, Yu Z. ^***

Bauman Moscow State Technical University, MSTU, 5, bldg. 1, 2-nd Baumanskaya str., Moscow, 105005, Russia

*e-mail: antt45@mail.ru
**e-mail: spm@bmstu.ru
***e-mail: yuzhaokai933@mail.ru

Abstract

The author derived a boundary condition considering energy dissipation near the three-phase contact line based on the Hamilton—Ostrogradsky principle. A numerical algorithm has been developed for calculating the damping factor due to energy dissipation near the three-phase contact line based on the finite element method. A variation formulation of the problem is derived from the of the motion equations linearization of the liquid relative to the equilibrium free surface. The area occupied by the fluid was discretized by the finite elements, and the eigenvalue problem, which solution represented a complex frequency, was obtained. In the work being presented, a pendulum and a spiral spring simulate the impact of mass forces and surface tension force respectfully, and the fluid viscosity is accounted for by the linear damper. The mechanical analog parameters are determined from the principle of dynamic similarity, eigen frequency, damping factor and kinetic energy of the fluid and its mechanical model. The article presents quantitative estimation of the capillary number Ca (viscous force and surface tension force interrelation), the Bond number B0 (mass forces and surface tension force interrelation) and the liquid filling factor β (liquid volume to the vessel concavity ratio) effect on the damping factor and eigen frequency of the capillary liquid oscillations. It follows from the studies that the capillarity number greatly affects the energy dissipation near the three-phase contact, and its value in the range of 10–100 leads to the greater value of the damping factor of the order of the damping coefficient on the vessel wall. The obtained results may be employed for the dynamics and stability studies of the rooster super stages, upper-stage rocketsand other spacecraft with the liquid containing cavities.

Keywords:

three-phase contact line, contact angle, low-viscosity fluid, complex frequency, mechanical model, finite element method

References

1. Case K., Parkinson W. Damping of surface waves in an incompressible liquid, Journal of Fluid Mechanics, 1957, vol. 2, no. 2, pp. 172‒184. DOI: https://doi.org/10.1017/S0022112057000051
2. Abramson H.N. The Dynamic Behavior of liquids in Moving Containers, NASA SP-106, 1966, 467 p.
3. Rabinovich B.I. Vvedenie v dinamiku raket-nositelei i kosmicheskikh apparatov (Introduction to the dynamics of launch vehicles and spacecraft), Moscow, Mashinostroenie, 1975, 416 p.
4. Mikishev G.N. Eksperimental’nye metody v dinamike kosmicheskikh apparatov (Experimental methods in the dynamics of spacecraft), Moscow, Mashinostroenie, 1978, 247 p.
5. Chernous’ko F.L. Dvizhenie tverdogo tela s polostyami, soderzhashchimi vyazkuyu zhidkost’ (The motion of a rigid body with cavities containing a viscous fluid), Moscow, Vychislitel’nyi tsentr AN SSSR, 1968, 232 p.
6. Bogoryad E.D. Dinamika vyazkoi zhidkosti so svobodnoi poverkhnost’yu (Dynamics of a viscous fluid with a free surface), Tomsk, Izd-vo Tomskogo universiteta, 1980, 101 p.
7. Miles J.W. Surface-wave damping in closed basins, Proceedings of the Royal Society of London, 1967, vol. 297, issue 1451, pp. 459‒475. URL: https://doi.org/10.1098/rspa.1967.0081
8. Henderson D.M, Miles J.W. Surface-wave damping in a circular cylinder with a fixed contact line, Journal of Fluid Mechanics, 1994, vol. 275, pp. 285‒299. URL: https://doi.org/10.1017/S0022112094002363
9. Wang W., Li J., Wang T. Modal analysis of liquid sloshing with different contact line boundary conditions using FEM, Journal of Sound and Vibration, 2008, vol. 317, issues 3–5, pp. 739–759. DOI: 10.1016/j.jsv.2008.03.070
10. Utsumi M. Slosh damping caused by friction work due to contact angle hysteresis, AIAA Journal, 2017, vol. 55, no. 1. pp. 1‒9. DOI: 10.2514/1.J055238
11. Dodge F.T., Kana D.D. Dynamics of liquid sloshing in upright and inverted bladdered tanks, Journal of fluids engineering, 1987, vol. 109, no. 1, pp. 58-63. URL: https://doi.org/10.1115/1.3242617
12. Kolesnikov K.S. Dinamika raket (Rocket dynamics), Moscow, Mashinostroenie, 2003, 520 p.
13. Li Q., Ma X., Wang T. Equivalent mechanical modal for liquid sloshing during draining, Acta Astronautica, 2011, vol. 68, issues 1-2, pp. 91-100. URL: https://doi.org/10.1016/j.actaastro.2010.06.052
14. Popov I.P. Trudy MAI, 2022, no. 126. URL: https://trudymai.ru/eng/published.php?ID=168987. DOI: 10.34759/trd-2022-126-01
15. Avduevskii V.S., Polezhaev V.I. Gidromekhanika i teplomassoobmen v nevesomosti (Hydromechanics and heat and mass transfer in weightlessness), Moscow, Nauka, 1982, 288 p.
16. Myshkis A.D., Babskii V.G., Zhukov M.Yu., Kopachevskii N.D., Slobozhanin L.A., Tyuptsov A.D. Metody resheniya zadachi gidromekhaniki dlya uslovii nevesomosti (Methods for solving the problem of hydromechanics for weightlessness conditions), Kiev, Naukova Dumka, 1992, 592 p.
17. Dodge F.T. The new «Dynamic behavior of liquids in moving containers», NASA SP-106, 2000, 202 p.
18. Ibrahim R.A. Liquid sloshing dynamics: theory and applications, Cambridge, Cambridge University Press, 2005, 948 p.
19. Aleksandrov L.G., Konstantinov S.B., Markov A.V., Platov I.V. Trudy MAI, 2022, no. 127. URL: https://trudymai.ru/eng/published.php?ID=170335. DOI: 10.34759/trd-2022-127-07
20. Yui Chzhaokai, Temnov A.N. Inzhenernyi zhurnal: nauka i innovatsii, 2021, no. 3, pp. 1-11. URL: http://dx.doi.org/10.18698/2308-6033-2021-3-2060
21. Yui Chzhaokai. Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika, 2022, no. 78, pp. 151–165. DOI: 10.17223/19988621/78/12
22. Yui Chzhaokai, Temnov A.N. Trudy MAI, 2022, no. 126. URL: https://trudymai.ru/eng/published.php?ID=168991. DOI: 10.34759/trd-2022-126-05.
23. Vil’ke V.G. Teoreticheskaya mekhanika (Theoretical mechanics), Saint Petersburg, Izd-vo «Lan’», 2003, 304 p.
24. Zenkevich O. The finite element method in engineering science, London, Mc-Graw-Hill, 1971, 521 p.
25. Bathe K.J. Finite element procedures. 2nd edition, Waterton, 2014, 1065 p.

Download