Abstract:
The three-dimensional, inviscid and viscous flow instability modes that appear on a solid body rotation flow in a finite-length, straight, circular pipe are analysed. This study is a direct extension of the Wang & Rusak [66] analysis of axisymmetric instabilities on inviscid swirling flows in a pipe. The linear stability equations are the same as those derived by Kelvin (also known as Sir William Thomson) [24]. However, we study a general mode of perturbation that satisfies the inlet, outlet and wall conditions of a flow in a finite-length pipe with a fixed in time and in space vortex generator ahead of it. This mode is different from the classical normal mode of perturbations. The eigenvalue problem for the growth rate and the shape of the perturbations for any azimuthal wave number m consists of a linear system of partial differential equations in terms of the axial and radial coordinates (x; r). The stability problem is solved numerically for all azimuthal wave numbers m. The computed growth rates and the related shapes of the various perturbation modes that appear in sequence as a function of the base flow swirl ratio (ω) and pipe length (L) are presented. In the inviscid flow case, the m = 1 modes are the first to become unstable as the swirl ratio is increased and dominate the perturbation's growth rates in a certain range of swirl levels. The m = 1 instability modes compete with the axisymmetric (m = 0) instability modes as the swirl ratio is further increased. In the viscous flow case, the viscous damping effects reduce the modes' growth rates. The neutral stability line is presented in a Reynolds number (Re) versus swirl ratio (ω) diagram, and it can be used to predict the first appearance of axisymmetric or spiral instabilities as a function of Re and L. We use the Reynolds-Orr equation to analyse the various production terms of the perturbation's kinetic energy, and establish that the elimination of the flow axial homogeneity at high swirl levels as the underlying physical mechanism that leads to flow exchange of stability and to the appearance of both spiral and axisymmetric instabilities. The viscous effects in the bulk have only a passive influence on the modes' shapes and growth rates. These effects decrease with the increase of Re. We show that the inviscid flow stability results are the inviscid-limit stability results of high-Re rotating flows. The stability of the Lamb-Oseen vortex with b = 4 in a finite-length pipe is studied. The study reveals that the dynamics of such flow is dominated by the axisymmetric perturbation mode.