Instability in pipe flow

Abstract

The long-puzzling, unphysical result that linear stability analyses lead to no transition in pipe flow, even at infinite Reynolds number, is ascribed to the use of stick boundary conditions, because they ignore the amplitude variations associated with the roughness of the wall. Once that length scale is introduced (here, crudely, through a corrugated pipe), linear stability analyses lead to stable vortex formation at low Reynolds number above a finite amplitude of the corrugation and unsteady flow at a higher Reynolds number, where indications are that the vortex dislodges. Remarkably, extrapolation to infinite Reynolds number of both of these transitions leads to a finite and nearly identical value of the amplitude, implying that below this amplitude, the vortex cannot form because the wall is too smooth and, hence, stick boundary results prevail.

Stability of steady flow through an axially corrugated pipe

“…4 shows the variation of critical Reynolds number with wall corrugation amplitude for non-axisymmetric disturbances as determined in the present study as well as the values reported by Cotrell et al for axisymmetric disturbances. 12 In the range of corrugation amplitude examined, nonaxisymmetric modes are the least stable.…”

“…They concluded that for disturbances with the same axial periodicity as the corrugation, axisymmetric disturbances were stable at Reynolds numbers up to approximately 800 (the highest in their study), and nonaxisymmetric instability arose via a Hopf bifurcation at Re c % 200 in azimuthal wavenumber k ¼ 1, leading to helical wavelike instabilities. The study of Cotrell et al 12 was primarily concerned with instabilities in the low corrugation amplitude limit, considered disturbances whose axial wavenumbers were in general incommensurate with that of the corrugation and concentrated on axisymmetric disturbances with k ¼ 0. Their analysis was carried out for corrugations of axial wavelength equal to the mean radius.…”

Computer experiments on the onset of turbulence

“…Also shown are the Kolmogorov scale and the thickness of the viscous sub-layer. the pipe diameter) values [5]. We could determine this disappearance at infinite Reynolds number by extrapolating the amplitude linearly with one over the Reynolds number at large Reynolds numbers.…”

“…By introducing corrugations, a very unrealistic roughness, we showed through linear stability analysis that for a given Reynolds number, turbulence does occur at rather large amplitudes relative to the viscous sub-layer and the instability completely disappears at also rather large (0.3% of FIGURE 1. Critical transition Reynolds number as obtained from linear stability analysis of a corrugated pipe [5] and corrugated channels [6], with indicated roughness shapes. Open blue circles represent direct simulations of transition in a pipe initiated by injecting fluid at the wall [7].…”

Development length of sinusoidally pulsating laminar pipe flows in moderate and high Reynolds number regimes

“…Notable among them deal with the stability analysis of flows inside corrugated tubes, where the corrugations represent the surface roughness (see for example, Cotrell et al, 2008;Loh and Blackburn, 2010). These studies clearly showed that the critical Reynolds number (Re crit ), below which the pipe flow remains laminar, is strongly dependent on the height of the roughness and Re crit ?…”