## DYNAMIC STABILITY OF A STRAIGHT PIPE CONVEYING PULSATILE FLOW UNDER THERMAL LOADS

**Authors:**Lolov D.S.^{1}, Lilkova-Markova S.V.^{1}**Affiliations:**- University of Architecture, Civil Engineering and Geodesy

**Issue:**No 2 (2023)**Pages:**5-10**Section:**ARTICLES**URL:**https://ered.pstu.ru/index.php/mechanics/article/view/3777**DOI:**https://doi.org/10.15593/perm.mech/2023.2.01- Cite item

# Abstract

Pipes conveying fluid are considered as a fundamental dynamical problem in the field of fluid- structure interaction. They are widely used in the petroleum industry, in nuclear engineering, aviation and aerospace, in nanostructures. This article investigates the effect of temperature load on the dynamic stability of a straight pipe conveying pulsatile flow. The fluid velocity is a harmonic function of time. The Galerkin method is applied for the solution of the differential equation of the transverse vibrations of the pipe. The differential equation is reduced to a first-order differential equation system. The system of differential equations is transformed and rewritten in a matrix form. The harmonic function of the fluid velocity allows the Floquet theory to be applied in order to investigate the dynamic stability of the system. The static scheme of the investigated pipe is a beam with restricted horizontal and vertical displacements at both of its ends. A numerical solution for a straight pipe conveying fluid with specified geometric and physical characteristics has been carried out. The temperature load and the constant fluid rate are considered as parameters of the problem. The results show that the temperature load affects the vibrational characteristics of the pipe, as well as its critical velocity.

#### Keywords

pipe, fluid, dynamic stability, thermal load, pulsatile flow, pipe, fluid, dynamic stability, thermal load, pulsatile flow.

# Full Text

Fluid conveying pipes find applications in a number of areas of engineering. They are widely used in the petroleum industry for transportation of oil and gas. Another broad use of them is in the transport of water. Pipelines are also primary structural parts in power plants, hydraulic systems, air-conditioners, refrigerators etc. Nanoscale tubes find application in nanophysics, nanobiology and nanomechanics as nanofluidic devices in nanocontainers for gas storage and nanopipes conveying fluid. The experiments at the nanoscale are difficult and expensive. That is why the continuum elastic models have been used to study the fluid-structure interaction. The carbon nanotubes are considered with Euler- and Timoshenkobeam models [1–9]. The flow of the fluid in the tube causes oscillations in it. The dynamic characteristics of the pipe’s oscillations depend on the velocity and the mass of the conveyed fluid. For pipes conveying fluid with a constant velocity it is known that the natural frequency of the pipe becomes lower when the velocity of the transported fluid increases. The velocity of the fluid corresponding to a natural frequency equal to zero is called critical velocity. At that point the system is at the edge of loss of stability. When the pipe conveys pulsatile flow, the pipe loses stability even though the mean velocity of the fluid is smaller than the critical velocity [10]. The research of the dynamic stability of pipes conveying fluid is branched into two directions: a) dynamic stability of pipes with a rectilinear axis [11–25] and b) dynamic stability of curved pipes [27–32]. The oscillations of a pipe with a flowing fluid, supported at both ends, were investigated in [36]. The global properties of the spectrum in dependence on fluid velocity, tube and fluid material densities, magnitude and direction of longitudinal force are established. In [37] the linear stability of elastic collapsible pipes with flowing fluid is investigated, in the case when the equilibrium configuration of the pipe is helical. The geometricvariational approach was applied to study the 3D dynamics of collapsible pipes. The dynamic stability of elastic membrane axisymmetric tubes filled with fluid was investigated in [38]. The considered fluid is non-viscous and incompressible. Thermal loads may induce excessive vibration in the system, leading to loss of stability. Therefore, analysis of the dynamic stability due to thermal loading is essential for the safe operation of the pipeline. The most common methods used for dynamic analysis of the pipes conveying fluid are the Transfer matrix method (TMM) and the Generalized differential quadrature method (GDQM). The both methods have significant advantage from the Finite element method (FEM). The conventional FEM can be very time consuming when it comes to investigation of a pipeline with a high number of spans. The order of the overall property matrices for the whole multispan pipeline increases with the number of spans. This is unlike the TMM in which the order of the overall transfer matrix is independent on number of spans and is kept unchanged. The GDQM approximates a derivative of a function in the partial differential equation of the lateral vibration of the pipe at any discrete point as a weighted sum of the function values at all discrete value at the domain. The main advantage of the method is its high convergence with a small number of grid points. The paper is structured as follows. First, it is presented the model of the pipe and the governing differential equation of its transverse vibration. The Galerkin method is employed to approach the solution of the problem. The Floquet theorem is applied to investigate the stability of the trivial solution. Finally, the obtained results from the numerical solution are presented and several important conclusions are summarized.# About the authors

### D. S. Lolov

University of Architecture, Civil Engineering and Geodesy

### Sv. V. Lilkova-Markova

University of Architecture, Civil Engineering and Geodesy

# References

- Belhadi A., Boukhalfa A., Belalia S.A. Free vibration modeling of single-walled carbon nanotubes using the differential quadrature method. Mathematical modeling of engineering problems. 2017, Vol. 4 (1), pp. 33-37.
- Reddy, C.D., Lu C., Rajendran S., Liew K.M. Free vibration analysis of fluid conveying single-walled carbon nanotubes. Applied Physics Letters. 2007, Vol. 90, pp. 133122.
- Yoon, J., Ru C., Mioduchowski A. Vibration and instability of carbon nanotubes conveying fluid. Composites Science and Technology. 2005, Vol.65 - pp. 1326 - 1336.
- Lolov D.S., Lilkova-Markova Sv. V. Dynamic stability of double-walled carbon nanotubes. Journal of the Serbian Society for Computational Mechanics. 2018, Vol.12(1), pp.1-8.
- Yoon J., Ru C.Q., Mioduchowski A, Flow-induced flutter instability of cantilever CNTs. Int. J. Solids Struct. 2006, Vol.43, pp. 3337–3349.
- Yoon J., Ru C.Q., Mioduchowski A, Vibration and instability of CNTs conveying fluid. Compos. Sci. Technol. 2005, Vol. 65, pp. 1326–1336.
- Wang X.Y., Wang X., Sheng G.G. The coupling vibration of fluid-filled carbon nanotubes. J. Phys. D: Appl. Phys. 2007, Vol.40, pp.2563–2572.
- Dong K., Wang X. Wave propagation in carbon nanotubes under shear deformation. Nanotechnology. 2006, Vol. 17, pp. 2773–2782.
- Dong K., Wang X., Sheng G.G. Wave dispersion characteristics in fluid-filled carbon nanotubes embedded in an elastic medium. Model. Simul. Mater. Sci. Eng. 2007, Vol.15, pp. 427–439.
- Jeong W., Soo Y., Jeong S.,Lee S., Yoo W. Stability Analysis of a pipe conveying periodically pulsating fluid using finite element method. JSME International Journal. 2006, Vol. 49, pp. 1116-1122.
- Paidoussis M. Fluid-structure interactions: Slender Structures and Axial flow, Academic Press, London, 1998.
- Paidoussis M., Issid N. Dynamic stability of pipes conveying fluid. Journal of Sound and Vibration. 1974, Vol. 33(3), pp. 267-284.
- Deng Q., Yang Z. Wave propagation in submerged pipe conveying fluid. Acta Mechanica Solida Sinica. 2019, Vol.32(4), pp. 483-498.
- Lolov D. S., Lilkova-Markova Sv. Dynamic stability of a fluid-immersed pipe conveying fluid and resting on a damped Winkler elastic foundation. Proceedings of XI International Conference Industrial Engineering and Environment Protection 2021 (IIZS 2021), October 7-8th, 2021, Zrenjanin, Serbia. pp.49-55.
- Hellum A., Mukherjee R., Hull A. Flutter instability of a fluid-conveying fluid-immerced pipe affixed to a rigid body. Journal of Fluids and Structures. 2011, Vol.27, pp. 1086-1096.
- Lin W., Qiao N. Vibration and stability of an axially moving beam immersed in fluid. International Journal of Solids and Structures. 2008, Vol. 45, pp. 1445-1457.
- Huang Q., Lin T., Safarpour M. Flow-induced vibration attenuation of a viscoelastic pipe conveying fluid under sinusoidal flow, using a nonlinear absorber. Mechanics Based design of Structures and Machines. 2020, Vol. 50, pp. 1673-1703.
- Liang F., Gao A., Yang X.D. Dynamical analysis of spinning functionally graded pipes conveying fluid with multiple spans. Applied Mathematical Modeling. 2020, Vol. 83, pp. 454-469.
- Askarian A.R., Permoon M.R., Shakouri M. Vibration analysis of pipes conveying fluid resting on a fractional Kelvin- Voigt viscoelastic foundation with general boundary conditions. International Journal of Mechanical Sciences. 2020. Vol. 179, pp. 105702.
- Cao J., Liu Y., Liu W. The effect of two cases of temperature distributions on vibration of fluid-conveying functionally graded thin-walled pipes. Journal of Strain Analysis. 2018, Vol. 53(5), pp. 324-331.
- Ameen K.A., Al-Dulaimi M.J., Hatem A.A. Experimental study of vibration on pipe conveying fluid at different end conditions for different fluid temperatures. Engineering and Technology Journal. 2019, Vol. 37, pp. 512-515.
- Li B., Wang Z., Jing L. Dynamic response of pipe conveying fluid with lateral moving supports. Shock and Vibration. 2018, Vol. 2018, pp. 1-17.
- Al-Waily M., Al-Baghdadi M., Al-Khayat R. Flow Velocity and Crack Angle Effect on Vibration and Flow Characterization for Pipe Induced Vibration. International Journal of Mechanical Mechatronics Engineering. 2017, Vol. 17(5), pp. 19-27.
- Siba M., Wahmahmood W., Zakinuaw M., Rasani R., Nassir M. Flow-induced vibrations in pipes: challengess and solutions – a review. Journal of Engineering Science and Technology. 2016, Vol. 11(3), pp. 362-382.
- Faal R., Derakhshan D. Flow-Induced Vibration of Pipeline on Elastic Support. Procedia Engineering. 2022, Vol. 14, pp. 2986-2993.
- Shiwen L., Ronghui H., Zhao Y., Jiarui W., Shuang C., Jianyong L., Yi L. Numerical Simulation Research on Flow-Induced Vibration Characteristics of Fluid-Conveying Pipe Network. Nuclear Power Engineering. 2022, Vol. 43(1), pp. 187-191.
- Jung D, Chung J. In-plane and out-of-plane motions of an extensible semi- circular pipe conveying fluid. Journal of Sound and Vibration, 2008, Vol.311(1), pp.408–420.
- Liu G., Li S., Karney B.Y. Vibration analysis of curved pipes conveying fluid. Proceedings of the ASME 2014 Pressure Vessels Piping Conference, Anaheim, California, USA. 2014, Vol.3.
- Lolov D., Lilkova-Markova Sv. Free vibrations out-ofplane of curved planar pipe conveying fluid in two cases of supporting. Annual of UACEG, Mathematics. Mechanics. 2006, Vol. 72(2), pp.123-128.
- Liang F., Yang X.-D., Bao R.-D., Zhang W. Frequency Analysis of Functionally Graded Curved Pipes Conveying Fluid. Advances in Materials Science and Engineering. 2016, Vol.2016, Article ID 7574216.
- Zhao Q., Sun Z. Flow-induced vibration of curved pipe conveying fluid by a new transfer matrix method. Engineering applications of computational applied mechanics. 2018, Vol. 12(1), pp. 780-790.
- Zhao Q., Sun Z. In-plane forced vibration of curved pipe conveying fluid by Green function method. Applied Mathematics and Mechanics (English Edition). 2017, Vol.38(10), pp. 1397-1414.
- Wu J., Shih P. The dynamic analysis of a multispan fluid-conveying pipe subjected to external load. Journal of sound and vibration. 2001, Vol.239(2), pp. 201-215.
- Chicone C. Ordinary Differential Equations with applications, Springer Science+ Business, 1999
- Inspeger Т., Horvath R. Pendulum with harmonic variation of the suspension point. Periodica Polytechnica. 2000, Vol. 44, pp. 39-46.
- Akulenko L.D., Georgievskii D.V., Nesterov S.V. Spectrum of Transverse Vibrations of a Pipeline Element under Longitudinal Load. Doklady Physics. 2016, Vol. 61 (3), pp. 129–132.
- Gay-Balmaz F., Georgievskii D., Putkaradze V. Stability of Helical Tubes Conveying Fluid. Journal of Fluids and Structtures. 2018. Vol 78 (2), pp. 146–174.
- Il’ichev A.T. Dynamics and spectral stability of solitonlike structures in fluid-filled membrane tubes. Russian Math. Surveys. 2020. Vol 75 (5), pp. 843–882