Aerospace Technic and Technology

The subject of this study is the unsteady flow of liquid in pipelines, which are part of the design of airplanes and helicopters. This name means, first of all, the phenomenon of a sharp increase in pressure in the pipeline, which is known as a hydraulic shock. Although we have already learned to deal with this phenomenon in some parts of the systems, in many structural elements (flexible pipelines), inside which the working pressure reaches several hundred atmospheres, this phenomenon is still quite dangerous. As you know, the best way to deal with an unwanted phenomenon is through theoretical study. To date, there has been a huge amount of work in the direction of hydraulic shock research. This article does not fully cover these studies. It is limited to references to reviews and relevant works. Because the phenomenon of hydraulic shock has a significantly nonlinear character, analytical solutions of systems of equations corresponding to the simplest models were unknown until recently. This work presents, as an overview, already known analytical solutions describing the process of shock wave propagation. Most importantly, new achievements are given, both for the inviscid approximation and for considering internal viscous friction. It is shown that the internal friction within the considered model is negligible almost everywhere, except for the thin shock layer. The asymptotic is proportional to the tangent function and inversely proportional to the square root of the product of the Reynolds number and the dimensionless parameter characterizing the convection effect. Convection of the velocity field significantly affects the distribution of characteristics in hydraulic shock. If the self-similar solutions that were obtained earlier have a power-law character for the velocity distribution in the shock wave, then the simultaneous consideration in the model of convection and friction on the pipeline walls (according to the Weisbach-Darcy model) made it possible to obtain a distribution in the form of an exponential function that decays with increasing distance from the shock wave front. In addition, the work includes an original approach to solving a nonlinear system of differential equations that describes the propagation of a shock wave without considering the friction on the walls. Analytical solutions were obtained in the form of a function of pressure versus the velocity of shock wave propagation. Research methods. This work uses purely theoretical approaches based on the use of well-known fluid flow models, methods of analytical solution of differential equations and their systems, asymptotic methods, derivation of self-model equations, and finding their solutions. Conclusions. Analytical solutions of systems of differential equations were obtained, which describe models of hydraulic shock without considering viscous effects. A comparison of the obtained results with the results of other studies is given.

plane; helicopter; structural elements; hydraulic shock (shock wave); stress; friction; surface deformation; fatigue

Krashanitsa, Yu. A., & Zheryakov, D. Yu. Aerodinamicheskiy profíl' v transzvukovom potoke gaza [Airfoil seсtion in the near-sonic flow of gas]. Aviacijno-kosmicna tehnika i tehnologia – Aerospace technic and technology, 2021, no. 2(170), pp. 20-27. DOI: 10.32620/aktt.2021.2.03. (in Russian).

Khorokhordin, A. O., Kravchenko, I. F., Mitrokhovych, M. M., Balalayeva, K. V., Balalayev, A. V. Metodyka ratsionalʹnoho formuvannya poverkhonʹ halʹmuvannya ploskoho nadzvukovoho vkhidnoho prystroyu [Method of rational formation of braking surfaces of a flat supersonic entry]. Aviacijno-kosmicna tehnika i tehnologia – Aerospace technic and technology, 2023, no. 4(190), pp. 28-34. DOI: 10.32620/aktt.2023.4sup2.03. (in Ukrainian).

Sivashenko, T. I. Dinamicheskiye i prochnostnyye kharakteristiki gibkikh metallicheskikh i ftoroplastovykh truboprovodov sistem letatel'nykh apparatov. Dis. ... kand. tekhn. nauk. [Dynamic and strength characteristics of flexible metal and fluoroplastic pipelines of aircraft system. Diss. … cand. tech. sci.]. Kyiv, 1972. 181 p. (in Russian).

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

Menabrea, L. F. Note sur les effects de choc de l’eau dans les conduits. C.R. Hebd. Seances Acad. Sci., Paris, Impr. de Mallet-Bachelier Publ., 1885, vol. 47, pp. 221-224. (in French).

Michaud, J. Coup de belier dans les conduits. Etude des moyens employes pour en atteneur less effects. Bulletin de la Société vaudoise des ingénieurs et des architectes, 1878, vol. 4(3.4), pp. 56-64, 65-77. DOI: 10.5169/seals-5907. (in French).

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

Allievi, L. Teoria generale del moto perturbato dell’acqua nei tubi in pressione (colpo d’ariete). Ann. Soc. Ing. Arch. Ithaliana, 1903, vol. 17, pp. 285-325. (in Italian).

Riemann, B. Ueber die Fortpflanzung Ebener Luftwellen von Endlicher Schwingungsweite (German Edition). Leopold Classic Library Publ., 2017. 32 p. (In German).

Ghidaoui, M. S., Zhau, M., McInnis, D. A., & Axworthy, D. H. A review of Water Hummer Theory and Practice. Applied Mechanics Reviews, 2005, vol. 58, iss. 1, pp. 49-76. DOI: 10.1115/1.1828050.

Kolpak, E. P., & Ivanov, S. I. Mathematical and Computer Modeling Vibrstion Protection System with Damper. Applied Mathematical Science, 2015, vol. 9, no. 78, pp. 3875-3885. DOI: 10.12988/ams.2015.53270.

Darcy, H. Recherches expérimentales relatives au mouvement de l'eau dans les tuyaux. Mallet-Bachelier, Paris, 1857. 268 p. Available at: https://archive.org/details/bub_gb_a4rcE52LFF0C/page/n11/mode/2up (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).

Tijsseling, A. S., Lambert, M. F., Simpson, A. R., Stephens, M. L., Vitkovsky, J. P., Bergand, & A. Skalak’s extentended theory of water hammer. Journal of Sound and Vibration, 2008, vol. 310, iss. 3, pp. 718-728. DOI: 10.1016/j.jsv.2007.10.037.

Skalak, R. An extention of the Theory of water hummer. Trans. ASME, 1956, vol. 78, no. 1, pp. 105-115. DOI: 10.1115/1.4013579.

Soloukhin, R. I. Udarnyye volny i detonatsiya v gazakh [Shock waves and detonation in gases]. Moscow, Fizmatlit Publ., 1963. 176 p. (in Russian).

Rankine, W. J. M. On the thermodynamic theory of waves of finite longitudinal disturbances. Philosophical Transactions of Royal Society of London. 1870, no. 160, pp. 277-288. Available at: https://www.jstor.org/stable/109061 (accessed 10 Dec. 2023).

Hugoniot, H. Memoir on the propagation of movements in bodies, especially perfect gases (first part). Journal de l’Ecole Polytechnique, 1887, no. 57, pp. 3-97. Available at: https://books.google.com.ua/books?id=wccAAAAAYAAJ&pg=PA3&redir_esc=y#v=onepage&q&f=false (accessed 10 Dec. 2023) (in French).

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. Processers, 2023, vol. 11, iss. 12, article no. 3305. DOI: 10.3390/pr11123305.

Adamkowski, A., Lewandowsky, M., & Lewandowsky, 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, T. W., & Leishear, R. A. When the Joukowsky Equation Does Not Predict Maximum Water Hammer Pressure. Journal of Pressure Vessel Technology, 2019, vol. 141, iss. 6, article no. 060801. 10 p. 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.