Theoretical study of the effect of unsteady viscous forces on inner- and annular-flow-induced instabilities of cylindrical shells

Abstract

A theoretical study is presented of the dynamical behaviour of a cylindrical shell coaxially located in a rigid cylindrical pipe, with viscous and incompressible fluid flow in the inner shell and the annulus. The fluid forces consist of two parts: (i) steady viscous forces representing the effects of upstream pressurization of the flow (to overcome frictional pressure drop) and skin friction on the shell surface, which are determined by using turbulent fully developed flow theory; (ii) unsteady viscous forces which are derived by means of linearized Navier-Stokes equations. Shell motion is described by Flügge's shell equations, modified to take the initial loading due to the steady viscous forces into account. A travelling wave solution is used to formulate the dynamical fluid-structure interaction problem. The objective is to investigate the effects of unsteady viscous forces on the dynamical behaviour and stability of the system in the presence and absence of steady forces. Calculations have been conducted with a steel shell conveying water with different gap-to-radius ratios g a i = 1 10 and 1 100 . First, the system is subjected to unsteady viscous forces only. The results are compared to those of inviscid theory. It is found that, for internal flow and annular flow for g a i = 1 10 , the effects of viscosity on the stability of the system are insignificant; however, for the smaller gap ( g a i = 1 100 these effects are more pronounced, rendering the system more stable. When both steady and unsteady viscous forces are applied, the results are quite different from the previous case. For annular flow, the system loses stability at much lower velocities for both gap systems. The loss of stability depends primarily on the steady viscous forces, at least for the parameters considered in the paper. The unsteady viscous forces affect only the frequencies of the system before it becomes unstable.