In-plane and out-of-plane dynamics of a curved pipe conveying pulsating fluid

Abstract

This paper investigates the in-plane and out-of-plane dynamics of a curved pipe conveying fluid. Considering the extensibility, von Karman nonlinearity, and pulsating flow, the governing equations are derived by the Newtonian method. First, according to the modified inextensible theory, only the out-of-plane vibration is investigated based on a Galerkin method for discretizing the partial differential equations. The instability regions of combination parametric resonance and principal parametric resonance are determined by using the method of multiple scales (MMS). Parametric studies are also performed. Then the differential quadrature method (DQM) is adopted to discretize the complete pipe model and the nonlinear dynamic equations are carried out numerically with a fourth-order Runge–Kutta technique. The nonlinear dynamic responses are presented to validate the out-of-plane instability analysis and to demonstrate the influence of von Karman geometric nonlinearity. Further, some numerical results obtained in this work are compared with previous experimental results, showing the validity of the theoretical model developed in this paper.

Appendix A

Expressions of nonlinear terms of the governing equations are listed below.

$$\begin{aligned} N_{1}& = - A \bigl( u''w' + u'w'' - \pi u'^{2} - \pi uu'' + \pi w'^{2}\\ &\quad {} + \pi ww'' - \pi^{2}u'w - \pi^{2}uw' \bigr) - A\biggl(\frac{1}{2}\pi u'^{2}\\ &\quad {} + \pi^{2}u'w + \frac{1}{2}\pi^{3}w^{2} + \frac{1}{2}\pi w'^{2} - \pi^{2}uw'\\ &\quad {} + \frac{1}{2}\pi^{3}u^{2} + \frac{1}{2}\pi v'^{2} + \frac{3}{2}u'^{2}u'' + 2\pi u'u''w \\ &\quad {}+ \pi u'^{2}w' + \frac{1}{2}\pi^{2}u''w^{2} + \pi^{2}u'w'w + \frac{1}{2}u''w'^{2}\\ &\quad {} + u'w'w'' - \pi uu''w' - \pi u'^{2}w' - \pi uu'w''\\ &\quad {} + \frac{1}{2} \pi^{2}u^{2}u'' + \pi^{2}uu'^{2} + \frac{1}{2}\pi u'^{2}w' + \pi u'u''w \\ &\quad {} + 2\pi^{2}u'ww' + \pi^{2}u''w^{2} + \frac{3}{2}\pi^{3}w^{2}w' + \frac{1}{2}\pi w'^{3}\\ &\quad {} + \pi ww'w'' - \pi^{2}uw'^{2} - \pi^{2}u'ww' - \pi^{2}uww''\\ &\quad {} + \frac{1}{2}\pi^{3}u^{2}w' + \pi^{3}uu'w + \frac{1}{2}u''v'^{2} + u'v'v'' \\ &\quad {}+ \frac{1}{2} \pi w'v'^{2} + \pi wv'v'' \biggr), \\ N_{2}& = - A \bigl( v'w'' - \pi v'u' + v''w' - \pi v''u \bigr) \\ &\quad {} - A\biggl(v'u'u'' + \pi v'u''w + \pi v'u'w' + \pi^{2}v'ww'\\ &\quad {} + v'w'w'' - \pi v'u'w' - \pi v'uw'' + \pi^{2}v'uu'\\ &\quad {} + v'^{2}v'' + \frac{1}{2}v''u'^{2} + \pi v''u'w + \frac{1}{2} \pi^{2}v''w^{2}\\ &\quad {} + \frac{1}{2}v''w'^{2} - \pi v''uw' + \frac{1}{2} \pi^{2}v''u^{2} + \frac{1}{2}v''v'^{2} \biggr), \\ N_{3} & = - A \bigl( u'w' - \pi uu' + \pi ww' - \pi^{2}uw \bigr) \\ &\quad {} + A\biggl(u'u'' + \pi u''w + \pi u'w' + \pi^{2}ww' + w'w''\\ &\quad {} - \pi u'w' - \pi uw'' + \pi^{2}uu' + v'v'' - \frac{1}{2}\pi u'^{3}\\ &\quad {} - \pi^{2}u'^{2}w - \frac{1}{2} \pi^{3}u'w^{2} - \frac{1}{2}\pi u'w'^{2} + \pi^{2}uu'w'\\ &\quad {} - \frac{1}{2}\pi^{3}u^{2}u' - \frac{1}{2}\pi^{2}u'^{2}w - \pi^{3}u'w^{2} - \frac{1}{2} \pi^{4}w^{3} \\ &\quad {}- \frac{1}{2}\pi^{2}ww'^{2} + \pi^{3}uww' - \frac{1}{2}\pi^{4}u^{2}w \\ &\quad {} - \frac{1}{2}\pi u'v'^{2} - \frac{1}{2}\pi^{2}wv'^{2}\biggr). \end{aligned}$$