Stability of steady flow through an axially corrugated pipe

Abstract

The linear stability of steady flow in pipes with circular cross-section and sinusoidal axial variation in diameter is studied by finding global eigenmodes with axial wavelength commensurate with that of the wall corrugation, chosen to be equal to one pipe mean radius. The maximum peak-to-peak height of corrugation considered is approximately 8% of the mean diameter. At low corrugation amplitude and at low Reynolds numbers, the base flow remains attached to the wall, while at larger amplitudes and Reynolds numbers, an axisymmetric separation bubble forms within the corrugation. For all Reynolds numbers considered, flows remain stable to axisymmetric perturbations, but become unstable to standing-wave modes of low azimuthal wavenumber, with critical Reynolds number first falling, then increasing with increasing corrugation height. Both attached and separated flows exhibit similar types of instability modes, which in the case of separated flow are most energetic near the reattachment line of the base flow. The leading instability modes consist of counter-rotating vortices situated near the pipe wall.The stability of fully developed flow in straight-walled pipes of circular section has been extensively studied and it is generally accepted 1-3 that the flow is linearly stable to infinitesimal perturbations, gives rise to moderate linear transient energy growth whose magnitude scales with the square of Reynolds number, 4 supports equilibrium (but unstable) travelling-wave states of finite amplitude, and becomes turbulent in practice at a range of bulk-flow Reynolds numbers Re ¼ U bulk D/ ¼ 4Q/(pD) starting at approximately 2000. The size of flow perturbations which trigger transition falls with Reynolds number, again for Reynolds numbers larger than approximately 2000. 5 The travelling waves are associated with longitudinal rolls and axial streaks and it has recently been suggested that the equilibrium states can be obtained as solutions of a nonlinear eigenproblem involving a coupling of these basic flow elements. 6 All real pipes have some degree of geometric imperfection, which may be randomly distributed (e.g., "commercial roughness"), or more organized, either by manufacture (e.g., machining or deliberately introduced during forming for the case of flexible segmented pipelines and ducts) or from scale deposition in process equipment. It is notable that over a large range of surface roughness, typically observed transition Reynolds numbers remain in the range 2000-3000, 7,8 suggesting that the basic mechanism of instability remains the same in the presence of wall roughness.Compared to the situation for flow perturbation in straight pipes, the effect of wall shape variation on pipe flow stability has received little systematic attention. Arguably, the simplest case for study is presented by a sinusoidal axial variation in diameter where the corrugation amplitude is described by a ¼ 0.5(D max À D min) /D mean. Experimental studies have mainly concentrated on comparatively large corrugation amplitudes a ' 0:3 À 0:...

Effects of Wavy Channel Entrance Design on Streamwise Counter-rotating Vortices: a Visualization Study

“…4(b). This formation of separation bubble is in conjunction with the study by Loh and Blackburn (2011). The structure was still noticeable at the second peak (χ = 3.47) before it became more indistinct as it travels further downstream prior to its breakdown.…”

On scaling pipe flows with sinusoidal transversely corrugated walls: analysis of data from the laminar to the low-Reynolds-number turbulent regime

“…The inset to figure 10 demonstrates that with increasing corrugation amplitude, a peak in the azimuthal energy spectrum begins to emerge at k o = 3. Loh & Blackburn (2011) found that for a corrugation wavelength of L m /D = 0.5, similar to the presently employed value of L m /D ≈ 0.419, steady axisymmetric flow in corrugated pipes became unstable to global modes with comparatively low azimuthal wavenumbers of k o = 3 and k o = 4. Hence, the emergence of this peak with increasing corrugation height at k o = 3 seems likely to be related to this instability mechanism.…”

“…Nikuradse's classic experimental data helps us to carry Loh & Blackburn (2011). It is interesting to observe that the onset of turbulence begins more rapidly with increasing Re D for the corrugated wall cases than for sand roughness.…”

“…We note that these flows may be unstable to non-axisymmetric disturbances. Stability analysis carried out by Loh & Blackburn (2011) for corrugated pipes with a similar corrugation wavelength to that employed here showed that the flow first became unstable to disturbances with azimuthal wavenumbers k o = 3, 4 and at bulk Reynolds numbers similar to the upper end of the range we have used.…”

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 ?…”