Aerospace Technic and Technology

The subject of this work is the development of a nonlinear model of the interaction of an unsteady fluid flow with a structure and finding analytical solutions for the system of equations that correspond to the specified model. The convection effect of the fluid velocity field was already considered in the previous works of the authors of this paper. These studies are devoted to the water hammer without considering the "flow-structure" interaction. This work expands the possibilities of modeling and considers four equations instead of two equations of the theory of the water hammer (equations of conservation of mass and momentum), two of which relate to the motion of particles of a solid body (pipes or structures). The novelty of this work lies in the consideration of the model that describes the interaction of the flow with the structure, the convection in the velocity field, and the effect, together with the friction of the fluid against the solid wall, on the dynamics of the shock pulse propagation process both in the fluid and in the solid body. It should be noted that the solution of the nonlinear system of differential equations as a whole is carried out by an analytical method, which makes it possible to obtain an exact (rather than numerical) solution of the problem. Since the effects of various factors should be evaluated by comparison with the main components, dimensionless equations containing six parameters (dimensionless combinations) were obtained in this study. Two of these parameters were named after scientists – Darcy and Weisbach (steady friction) and Bruno (unsteady friction).  Particular cases of the general (full) model were considered, and the effects of various factors on the dynamics of the interaction of the flow with the structure during the propagation of the shock pulse were determined. Research methods are purely theoretical. The concepts of a self-similar equation and a system of equations, balances of forces acting on particles of a fluid and a solid body, and a standard method of reducing a system of equations to a single equivalent equation are used. Conclusions. An extended model of the interaction between the unsteady fluid flow and the structure is proposed. The transition to a self-similar variable makes it possible to solve a nonlinear system of differential equations and obtain an analytical (exact) solution. The functions of longitudinal stress in a solid body, pressure disturbance, and velocity of motion of particles in a solid body (pipe) are linearly expressed by the velocity of shock pulse propagation in the fluid. It should also be noted that the results for the particular case of the linear model completely agree with the already known ones. The advantage of using a self-similar solution is that it is easy to obtain. The results of previous studies on the water hammer problem were qualitatively consistent. As the fluid viscosity increases, the shock pulse domain becomes more concentrative.

aircraft; helicopter; incompressible (droplet) fluid; flow-structure interaction; water hammer; stress; surface deformation; fatigue

Su, H., Sheng, L., Zhao, S., Lu, C., Zhu, R., Chen, Y., & Fu, Q. Water Hammer Characteristics and Component Fatigue Analysis of the Essential Service Water System in Nuclear Power Plants. Processes, 2023, vol. 11, iss. 12, article no. 3305. DOI: 10.3390/pr11123305.

Adamkowski, A., Lewandowski, M., & Lewandowski, S. Fatigue life analysis of hydropower pipelines using the analytical model of stress concentration in welded joints with angular distortions and considering the influence of water hammer damping. Thin-Walled Structures, 2021, vol. 159, article no. 107350. pp. 1-12. DOI: 10.1016/j.tws.2020.107350.

Walters, W. T., & Leishear, R. A. When the Joukowsky Equation Does Not Predict Maximum Water Hammer Pressures. Journal of Pressure Vessel Technology, vol. 141, iss. 6, article no. 060801, pp. 1-10. DOI: 10.1115/1.4044603.

Leishear, R. A. Water hammer and Fatigue Corrosion –I –A Piping System Failure Analysis. Leishear Engineering, LLC, 2023, pp. 1-19. DOI: 10.13140/RG.2.2.19541.09449.

Allievi, L. B. Teoria generale del moto perturbato dell’acqua nei tubi in pressione (colpo d’ariete). Annali della Società degli ingegneri e degli architetti italiani, 1903, vol. 17, pp. 285-325. Available at: https://digit.biblio.polito.it/1119/ (accessed Jan. 10 2024).

Joukowski, N. E. Memories of Imperial Academy Society of St. Petrburg, vol. 9, iss. 5. (Russian translated by O. Simin 1904) Proc. Amer. Water Assoc., 1898, vol. 24, pp. 341-424.

Skalak, R. An extension of the theory of water hammer. Water Power, 1955, no. 7, pp. 458-462.

Skalak, R. An extension of the theory of water hammer. Water Power, 1956, no. 8, pp. 17-22.

Tijsseling, A. S. Exact solution of linear hyperbolic four-equation system in axial liquid-pipe vibration. Journal of Fluids and Structures, 2003, vol. 18, iss. 2, pp. 179-196. DOI: 10.1016/j.jfluidstructs.2003.07.001.

Wiggert, D. C., & Tijsseling, A. S. Fluid transient and fluid-structure interaction in flexible liquid-filled piping. Applied Mechanics Reviews, 2001, vol. 54, iss. 5, pp. 455-481. DOI: 10.1115/1.1404122.

Li, Q. S., Asce, M., Yang, K., & Zhang, L. Analytical solution for Fluid-Stuctures interaction in liquid-filled pipes subjected to impact-induced water hammer. Journal of Engineering Mechanics, 2003, vol. 129, iss. 12, pp. 1408-1417. DOI: 10.1061/(ASCE)0733-9399(2003)129:12(1408).

Lukianov, P. V., Syvashenko, T. I., & Yakymenko, B. M. Udarna khvylya v ridyni, shcho znakhodytʹsya v pruzhniy tsylindrychniy obolontsi neskinchennoyi dovzhyny [Shock wave in a liquid located in an elastic cylindrical shell of infinite length]. Promyslova hidravlika і pnevmatyka – Industrial hydraulics and pneumatics, 2019, vol. 2(64), pp. 38-46.

Lukianov, P. V., & Pavlova, K. S. Unsteady flow of droplet liquid in hydraulic systems of aircraft and helicopters: models and analytical solutions. Aviacijno-kosmicna tehnika i tehnologia – Aerospace technic and technology, 2024, no. 1, pp. 32-42. DOI: 10.32620/aktt.2024.1.03.

Lukianov, P. V., & Pavlova, K. S. Unsteady flow in bubble liquid in hydraulic system of aircraft snd helicopters. Aviacijno-kosmicna tehnika i tehnologia – Aerospace technic and technology, 2024, no. 2, pp. 4-14. DOI: 10.32620/aktt.2024.2.01.

Darcy, H. Recherches expérimentales relatives au mouvement de l'eau dans les tuyaux. Mallet- Bachelier, Paris, 1857. 268 p. Available at: https://search.worldcat.org/title/Recherches-experimentales-relatives-au-mouvement-de-l'eau-dans-les-tuyaux/oclc/25608493 (accessed 10 Dec. 2023) (in French).

Weisbach, J. Lehrbuch der Ingenieur- und Maschinen-Mechanik, Vol. 1. Theoretische Mechanik, Braunschweig, Vieweg Publ., 1855. 946 p. Available at: https://www.digitale-sammlungen.de/de/view/bsb10081227 (accessed 10 Dec. 2023) (In German).

Bergant, A., Simpson, A. R., & Vitkovsky, J. Developments in unsteady pipe flow friction modeling. Journal of Hydraulic Research, 2001, vol. 39, iss. 3, pp. 249-257. DOI: 10.1080/00221680109499828.

Zielke, W. Frequency-Dependent Friction in Transient Pipe Flow. Journal of Basic Engineering, 1967, vol. 90, iss. 1, pp. 109-115. DOI: 10.1115/1.3605049.

Riemann, B. Über Die Fortpflanzung Ebener Luftwellen von Endlicher Schwingungsweite (German Edition). Leopold Classic Library Publ., 2017. 32 p. (In German). Available at: https://www.emis.de/classics/Riemann/Welle.pdf. (accessed 10 Dec. 2023) (In German).

Korn, G. A., & Korn, T. M. Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review (Dover Civil and Mechanical Engineering). Dover Publications; Revised edition, 2000. 1152 p. ISBN: 978-0486411477.