Hostname: page-component-8448b6f56d-m8qmq Total loading time: 0 Render date: 2024-04-16T12:05:56.224Z Has data issue: false hasContentIssue false
  • English
  • Français

Transition to turbulence in duct flow

Published online by Cambridge University Press:  17 January 2008

DAMIEN BIAU ,
HOUSSAM SOUEID  and
ALESSANDRO BOTTARO
DAMIEN BIAU
Affiliation:
Università di Genova, DICAT, Via Montallegro 1, 16145 Genova, Italy
HOUSSAM SOUEID
Affiliation:
Università di Genova, DICAT, Via Montallegro 1, 16145 Genova, Italy
ALESSANDRO BOTTARO
Affiliation:
Università di Genova, DICAT, Via Montallegro 1, 16145 Genova, Italy
Get access Rights & Permissions [Opens in a new window]

Abstract

The transition of the flow in a duct of square cross-section is studied. Like in the similar case of the pipe flow, the motion is linearly stable for all Reynolds numbers; this flow is thus a good candidate to investigate the ‘bypass’ path to turbulence. Initially the so-called ‘linear optimal perturbation problem’ is formulated and solved, yielding optimal disturbances in the form of longitudinal vortices. Such optimals, however, fail to elicit a significant response from the system in the nonlinear regime. Thus, streamwise-inhomogeneous sub-optimal disturbances are focused upon; nonlinear quadratic interactions are immediately caused by such initial perturbations and an unstable streamwise-homogeneous large-amplitude mode rapidly emerges. The subsequent evolution of the flow, at a value of the Reynolds number at the boundary between fully developed turbulence and relaminarization, shows the alternance of patterns with two pairs of large-scale vortices near opposing parallel walls. Such edge states bear a resemblance to optimal disturbances.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 596 , 25 January 2008 , pp. 133 - 142
Copyright
Copyright © Cambridge University Press 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Artuso, R., Aurell, E. & Cvitanović, P. 1990 Recycling of strange sets: I. Cycle expansion. Nonlinearity 3, 325–359.Google Scholar
Ben-Dov, G. & Cohen, J. 2007 Instability of optimal non-axisymmetric base-flow deviations in pipe Poiseuille flow. J. Fluid Mech., 588, 189–215.Google Scholar
Bergstrom, L. 1993 Optimal growth of small disturbances in pipe Poiseuille flow. Phys. Fluids A 5, 2710–2720.CrossRefGoogle Scholar
Biau, D. & Bottaro, A. 2004 Transient growth and minimal defects: Two possible initial paths of transition to turbulence in plane shear flows. Phys. Fluids 16, 3515–3529.CrossRefGoogle Scholar
Boberg, L. & Brosa, U. 1988 Onset of turbulence in a pipe. Z. Naturforsch. 43, 697–726.CrossRefGoogle Scholar
Botella, O. 1997 On the solution of the Navier-Stokes equations using Chebyshev projection schemes with third-order accuracy in time. Computers Fluids 26, 107–116.CrossRefGoogle Scholar
Bottaro, A., Corbett, P. & Luchini, P. 2003 The effect of base flow variation on flow stability. J. Fluid Mech. 476, 293–302.CrossRefGoogle Scholar
Bottaro, A., Soueid, H. & Galletti, B. 2006 Formation of secondary vortices in turbulent square-duct flow. AIAA J. 44, 803–811.CrossRefGoogle Scholar
Corbett, P. & Bottaro, A. 2000 Optimal perturbations for boundary layers subject to stream-wise pressure gradient. Phys. Fluids 12, 120–130.CrossRefGoogle Scholar
Eckhardt, B., Schneider, T. M., Hof, B. & Westerweel, J. 2007 Turbulence transition in pipe flow. Annu. Rev. Fluid Mech. 39, 447–468.CrossRefGoogle Scholar
Faisst, H. & Eckhardt, B. 2003 Traveling waves in pipe flow. Phys. Rev. Lett. 91, 224502.Google Scholar
Galletti, B. & Bottaro, A. 2004 Large-scale secondary structures in duct flow. J. Fluid Mech. 512, 85–94.CrossRefGoogle Scholar
Gavarini, M. I., Bottaro, A. & Nieuwstadt, F. T. M. 2004 The initial stage of transition in pipe flow: Role of optimal base flow distortions. J. Fluid Mech. 517, 131–165.Google Scholar
Gavrilakis, S. 1992 Numerical simulation of low-Reynolds-number turbulent flow through a straight square duct. J. Fluid Mech. 244, 101–129.CrossRefGoogle Scholar
Gill, A. E. 1965 On the behaviour of small disturbances to Poiseuille flow in a circular pipe. J. Fluid Mech. 21, 145–172.CrossRefGoogle Scholar
Hof, B., van Doorne, C. W. H., Westerweel, J. & Nieuwstadt, F. T. M. 2005 Turbulence regeneration in pipe flow at moderate Reynolds numbers. Phys. Rev. Lett. 95, 214502.Google Scholar
Hof, B., van Doorne, C. W. H., Westerweel, J., Nieuwstadt, F. T. M., Faisst, H., Eckhardt, B., Wedin, H., Kerswell, R. R. & Waleffe, F. 2004 Experimental observation of nonlinear traveling waves in turbulent pipe flow. Science 305, 1594–1598.CrossRefGoogle ScholarPubMed
Hof, B, Westerweel, J., Schneider, T. M. & Eckhardt, B. 2006 Finite lifetime of turbulence in shear flows. Nature 443, 59–62.CrossRefGoogle ScholarPubMed
Jones, O. C. 1976 An improvement in the calculation of turbulent friction in rectangular ducts. Trans. ASME: J. Fluids Engng 98, 173–181.Google Scholar
Kerswell, R. R. 2005 Recent progress in understanding the transition to turbulence in a pipe. Nonlinearity 18, R17–R44.Google Scholar
Landahl, M. T. 1980 A note on an algebraic instability of inviscid parallel shear flows. J. Fluid Mech. 98, 243–251.CrossRefGoogle Scholar
Nagata, M. 1990 Three-dimensional finite amplitude solutions in plane Couette flow. J. Fluid Mech. 217, 519–527.Google Scholar
Nagata, M. 1997 Three-dimensional traveling-wave solutions in plane Couette flow. Phys. Rev. E 55, 2023.CrossRefGoogle Scholar
Peixinho, J. & Mullin, T. 2006 Decay of turbulence in duct flow. Phys. Rev. Lett. 96, 094501.CrossRefGoogle Scholar
Pringle, C. & Kerswell, R. R. 2007 Asymmetric, helical and mirror-symmetric traveling waves in pipe flow. Phys. Rev. Lett. 99, 074502.Google Scholar
Salwen, H., Cotton, F. W. & Grosch, C. E. 1980 Linear stability of Poiseuille flow in a circular pipe. J. Fluid Mech. 98, 273–284.CrossRefGoogle Scholar
Schmid, P. J. & Henningson, D. S. 1992 A new mechanism for rapid transition involving a pair of oblique waves. Phys. Fluids A-4, 1986–1989.CrossRefGoogle Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.Google Scholar
Tatsumi, T. & Yoshimura, T. 1990 Stability of the laminar flow in a rectangular duct. J. Fluid Mech. 212, 437–449.CrossRefGoogle Scholar
Uhlmann, M., Pinelli, A., Kawahara, G. & Sekimoto, A. 2007 Marginally turbulent flow in a square duct. J. Fluid Mech. 588, 153–162.Google Scholar
Waleffe, F. 1997 On a self-sustaining process in shear flows. Phys. Fluids 9, 883–900.Google Scholar
Waleffe, F. 1998 Three-dimensional states in plane shear flow. Phys. Rev. Lett. 81, 4140–4143.CrossRefGoogle Scholar
Waleffe, F. 2003 Homotopy of exact coherent structures in plane shear flows. Phys. Fluids 15, 1517–1534.CrossRefGoogle Scholar
Wedin, H. & Kerswell, R. R. 2004 Exact coherent structures in pipe flow: travelling wave solutions. J. Fluid Mech. 508, 333–371.Google Scholar