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

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.

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

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