Geodesic
Linear and Nonlinear Development of Bending Perturbations in a Fluid-Conveying Pipe with Variable Elastic Properties
Informatics and Automation, Modern Methods of Mechanics, Tome 322 (2023), pp. 10-23.

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We consider bending vibrations of a fluid-conveying pipe resting on an elastic foundation with nonuniform elasticity coefficient. Previously A. G. Kulikovskii showed analytically that the elasticity parameters can be distributed in such a way that at every point the system is either locally stable or convectively unstable. In this case, despite the absence of local absolute instability, there exists a global growing mode whose formation is associated with the points of internal reflection of waves. In the present paper, we perform a numerical simulation of the development of the initial perturbation in such a system. In the linear formulation we demonstrate how the perturbation is transformed into a growing eigenmode after a series of reflections and passages through a region of local instability. In the nonlinear formulation, where the nonlinear tension of the pipe is taken into account within the von Kármán model, we show that the perturbation growth is limited; in this case the vibrations acquire a quasi-chaotic character but do not leave the region bounded by the internal reflection points determined by the linearized problem.
Keywords: absolute/convective instability, global instability, internal reflection, development of perturbations, hydroelasticity. 
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K. E. Abdul'manov; V. V. Vedeneev. Linear and Nonlinear Development of Bending Perturbations in a Fluid-Conveying Pipe with Variable Elastic Properties. Informatics and Automation, Modern Methods of Mechanics, Tome 322 (2023), pp. 10-23. http://geodesic.mathdoc.fr/item/TRSPY_2023_322_a1/

[1] Bakhvalov N.S., Kornev A.A., Chizhonkov E.V., Chislennye metody: Resheniya zadach i uprazhneniya, Drofa, M., 2009

[2] Chomaz J.-M., Huerre P., Redekopp L.G., “A frequency selection criterion in spatially developing flows”, Stud. Appl. Math., 84:2 (1991), 119–144 | DOI | MR | Zbl

[3] Coenen W., Lesshafft L., Garnaud X., Sevilla A., “Global instability of low-density jets”, J. Fluid Mech., 820 (2017), 187–207 | DOI | MR | Zbl

[4] Couairon A., Chomaz J.-M., “Fully nonlinear global modes in slowly varying flows”, Phys. Fluids, 11:12 (1999), 3688–3703 | DOI | MR | Zbl

[5] V. I. Feodos'ev, “On the vibrations and stability of a pipe with liquid flowing through it”, Inzh. Sb., 10 (1951), 169–170

[6] A. G. Kulikovskii, “On the stability loss of weakly non-uniform flows in extended regions. The formation of transverse oscillations of a tube conveying a fluid”, J. Appl. Math. Mech., 57:5 (1993), 851–856 | DOI | MR | Zbl

[7] Kulikovskii A.G., Shikina I.S., “Ob izgibnykh kolebaniyakh dlinnoi truby, zapolnennoi dvizhuscheisya zhidkostyu”, Izv. AN ArmSSR. Mekhanika, 41:1 (1988), 31–39 | MR | Zbl

[8] Le Dizès S., Huerre P., Chomaz J.M., Monkewitz P.A., “Linear global modes in spatially developing media”, Philos. Trans. R. Soc. London A, 354:1705 (1996), 169–212 | DOI | Zbl

[9] Monkewitz P.A., “The absolute and convective nature of instability in two-dimensional wakes at low Reynolds numbers”, Phys. Fluids, 31:5 (1988), 999–1006 | DOI

[10] Pier B., “On the frequency selection of finite-amplitude vortex shedding in the cylinder wake”, J. Fluid Mech., 458 (2002), 407–417 | DOI | Zbl

[11] Pier B., Huerre P., “Fully nonlinear global modes in spatially developing media”, Physica D, 97:1–3 (1996), 206–222 | DOI

[12] Pier B., Huerre P., Chomaz J.-M., “Bifurcation to fully nonlinear synchronized structures in slowly varying media”, Physica D, 148:1–2 (2001), 49–96 | DOI | MR | Zbl

[13] Pier B., Huerre P., Chomaz J.-M., Couairon A., “Steep nonlinear global modes in spatially developing media”, Phys. Fluids, 10:10 (1998), 2433–2435 | DOI | MR | Zbl

[14] Shishaeva A., Aksenov A., Vedeneev V., “The effect of external perturbations on nonlinear panel flutter at low supersonic speed”, J. Fluids Struct., 111 (2022), 103570 | DOI

[15] V. V. Vedeneev, “Limit oscillatory cycles in the single mode flutter of a plate”, J. Appl. Math. Mech., 77:3 (2013), 257–267 | DOI | MR | Zbl

[16] V. V. Vedeneev and A. B. Poroshina, “Stability of an elastic tube conveying a Non-Newtonian fluid and having a locally weakened section”, Proc. Steklov Inst. Math., 300 (2018), 34–55 | DOI | DOI | MR | Zbl

[17] Volmir A.S., Nelineinaya dinamika plastinok i obolochek, Nauka, M., 1972