The onset of absolute instability of rotating Hagen–Poiseuille flow: A spatial stability analysis

Abstract

A spatial, viscous stability analysis of Poiseuille pipe flow with superimposed solid body rotation is considered. For each value of the swirl parameter ͑inverse Rossby number͒ LϾ0, there exists a critical Reynolds number Re c (L) above which the flow first becomes convectively unstable to nonaxisymmetric disturbances with azimuthal wave number nϭϪ1. This neutral stability curve confirms previous temporal stability analyses. From this spatial stability analysis, we propose here a relatively simple procedure to look for the onset of absolute instability that satisfies the so-called Briggs-Bers criterion. We find that, for perturbations with nϭϪ1, the flow first becomes absolutely unstable above another critical Reynolds number Re t (L)ϾRe c (L), provided that LϾ0.38, with Re t →Re c as L→ϱ. Other values of the azimuthal wave number n are also considered. For Re ϾRe t (L), the disturbances grow both upstream and downstream of the source, and the spatial stability analysis becomes inappropriate. However, for ReϽRe t , the spatial analysis provides a useful description on how convectively unstable perturbations become absolutely unstable in this kind of flow.

A numerical method for the study of nonlinear stability of axisymmetric flows based on the vector potential

“…From it we obtain a non-dimensional period T N ' 8:894, which yields a frequency x N ¼ 2p=T N ' 0:7065. This value practically coincide with that corresponding to the most unstable frequency for the most unstable mode n ¼ À1 obtained from the linear stability theory for this case Re h ¼ 100 and Re ¼ 100 [17]: x S ' 0:7098. The agreement is also very good for the wavenumber: From Fig.…”

“…2(d)), without distorting or affecting them in any appreciable form. The lower neutral curve for convective instability ðn ¼ À1Þ was first obtained by Mackrodt [16]; all the curves are taken from [17]. The following case Re ¼ 100 is convectively unstable (see Fig.…”

“…To check the accuracy of the method we apply it to the flow in a rotating pipe, for which very abundant information is available in the literature on linear stability, nonlinear travelling waves, and numerical simulations [15][16][17][18][19][20][21]. In particular, Sanmiguel-Rojas and Fernandez-Feria [21] addresses the same problem but using a numerical technique based in primitive variables ðp; vÞ and Dirichlet boundary conditions for the pressure [22].…”

Impact of rotating and fixed nozzles on vortex breakdown in compressible swirling jet flows

“…For the nozzle at rest, helical instabilities are expected to grow within the nozzle wall boundary layer due to shear and centrifugal instability mechanisms [40], and a transition to turbulence takes place already for relatively short nozzles. For the rotating nozzle setup, the boundary layer at the inner nozzle wall is comparably stable and instabilities grow substantially slower in the downstream direction in accordance with [41]. The nozzle flow is expected to stay laminar due to its comparably short length, cf.…”

Nonparallel local spatial stability analysis of pipe entrance swirling flows

“…[9][10][11][12] It was found there that the most unstable disturbance corresponded to the mode with azimuthal wave number nϭϪ1. Spatial stability analyses 13,14 of the same basic flow have shown that the flow was convectively unstable under nonaxisymmetric perturbations; the most unstable mode being precisely the nϭϪ1 one previously found by temporal stability analyses. In addition, a transition from convective to absolute instability was found for sufficiently large values of the Reynolds number and swirl strength.…”