Instability modes on a solid-body-rotation flow in a finite-length pipe

Chunjuan Feng,Shixiao Wang,Zvi Rusak,Feng Liu

doi:10.1063/1.4993643

Abstract

Numerical solutions of the incompressible Navier-Stokes equations are obtained to study the time evolution of both axisymmetric and three-dimensional perturbations to a base solid-body-rotation flow in a finite-length pipe with non-periodic boundary conditions imposed at the pipe inlet and outlet. It is found that for a given Reynolds number there exists a critical swirl number beyond which the initial perturbations grow, in contrast to the solid-body rotation flow in an infinitely-long pipe or a finite-length pipe with periodic inlet and exit boundary conditions for which the classical Kelvin analysis and Rayleigh stability criterion affirm neutrally stable for all levels of swirl. This paper uncovers for the first time the detailed evolution of the perturbations in both the axisymmetric and three-dimensional situations. The computations reveal a linear growth stage of the perturbations with a constant growth rate after a brief initial period of decay of the imposed initial perturbations. The fastest growing axisymmetric and three-dimensional instability modes and the associated growth rates are identified numerically for the first time. The computations show that the critical swirl number increases and the growth rate of instability decreases at the same swirl number with decreasing Reynolds number. The growth rate of the axisymmetric mode at high Reynolds number agrees well with previous stability theory for inviscid flow. More importantly, three-dimensional simulations uncover that the most unstable mode is the spiral type m = 1 mode, which appears at a lower critical swirl number than that for the onset of the axisymmetric mode. This spiral mode grows faster than the unstable axisymmetric mode at the same swirl. Moreover, the computations reveal that after the linear growing stage of the perturbation the flow continues to evolve nonlinearly to a saturated axisymmetric vortex breakdown state.

Highlights

The dynamics of the swirling flow in a pipe at high levels of rotation is dominated by flow instabilities and the vortex breakdown phenomena
We note that a translational invariance property exists in the above studies of a flow in an infinitely-long straight pipe whereas such a translational invariance does not exist in real experimental apparatuses in which breakdown is studied
The stability and global analyses of Wang and Rusak[17,18] reveal that the solid-body-rotation flow in a circular pipe of a finite length exhibits an axisymmetric instability when non-periodic boundary conditions are applied at the inlet and exit of the pipe

Summary

Introduction

The dynamics of the swirling flow in a pipe at high levels of rotation is dominated by flow instabilities and the vortex breakdown phenomena (see Sarpkaya,[1,2] Faler and Leibovich,[3] Leibovich,[4] Brucker and Althaus,[5] Mattner et al.,[6] Liang and Maxworthy,[7] and Dennis et al.[8]). Several theories were advanced over the years to explain various aspects or mechanisms of the vortex breakdown phenomena, including the studies by Benjamin,[9] Randall and Leibovich[10] and Leibovich and Kribus,[11] who investigated the dynamics of standing or traveling nonlinear waves on swirling flows in an infinitely-long pipe. We note that a translational invariance property exists in the above studies of a flow in an infinitely-long straight pipe whereas such a translational invariance does not exist in real experimental apparatuses in which breakdown is studied.

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Methods

Conclusion