Hostname: page-component-848d4c4894-p2v8j Total loading time: 0 Render date: 2024-05-02T15:57:02.425Z Has data issue: false hasContentIssue false
  • English
  • Français

On the stability of plane Couette–Poiseuille flow with uniform crossflow

Published online by Cambridge University Press:  02 June 2010

ANIRBAN GUHA  and
IAN A. FRIGAARD
ANIRBAN GUHA*
Affiliation:
Institute of Applied Mathematics, University of British Columbia, 6356 Agricultural Road, Vancouver, BC, V6T 1Z2, Canada Department of Civil Engineering, University of British Columbia, 2002-6250 Applied Science Lane, Vancouver, BC, V6T 1Z4, Canada
IAN A. FRIGAARD
Affiliation:
Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, BC, V6T 1Z2, Canada Department of Mechanical Engineering, University of British Columbia, 2054-6250 Applied Science Lane, Vancouver, BC, V6T 1Z4, Canada
*
†Email address for correspondence: aguha@interchange.ubc.ca
Get access Rights & Permissions [Opens in a new window]

Abstract

We present a detailed study of the linear stability of the plane Couette–Poiseuille flow in the presence of a crossflow. The base flow is characterized by the crossflow Reynolds number Rinj and the dimensionless wall velocity k. Squire's transformation may be applied to the linear stability equations and we therefore consider two-dimensional (spanwise-independent) perturbations. Corresponding to each dimensionless wall velocity, k ∈ [0, 1], two ranges of Rinj exist where unconditional stability is observed. In the lower range of Rinj, for modest k we have a stabilization of long wavelengths leading to a cutoff Rinj. This lower cutoff results from skewing of the velocity profile away from a Poiseuille profile, shifting of the critical layers and the gradual decrease of energy production. Crossflow stabilization and Couette stabilization appear to act via very similar mechanisms in this range, leading to the potential for a robust compensatory design of flow stabilization using either mechanism. As Rinj is increased, we see first destabilization and then stabilization at very large Rinj. The instability is again a long-wavelength mechanism. An analysis of the eigenspectrum suggests the cause of instability is due to resonant interactions of Tollmien–Schlichting waves. A linear energy analysis reveals that in this range the Reynolds stress becomes amplified, the critical layer is irrelevant and viscous dissipation is completely dominated by the energy production/negation, which approximately balances at criticality. The stabilization at very large Rinj appears to be due to decay in energy production, which diminishes like Rinj−1. Our study is limited to two-dimensional, spanwise-independent perturbations.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 656 , 10 August 2010 , pp. 417 - 447
Copyright
Copyright © Cambridge University Press 2010

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

REFERENCES

Baines, P. G., Majumdar, S. J. & Mitsudera, H. 1996 The mechanics of the Tollmien–Schlichting wave. J. Fluid Mech. 312, 107–124.CrossRefGoogle Scholar
Berkowitz, B. 2002 Characterizing flow and transport in fractured geological media: a review. Adv. Water Resour. 25, 861–884.CrossRefGoogle Scholar
Chapman, S. J. 2002 Subcritical transition in channel flows. J. Fluid Mech. 451, 35–97.Google Scholar
Cowley, S. J. & Smith, F. T. 1985. On the stability of Poiseuille–Couette flow: a bifurcation from infinity. J. Fluid Mech. 156, 83–100.Google Scholar
Eckhardt, B., Schneider, T., Hof, B. & Westerweel, J. 1998 Turbulence transition in pipe flow. Annu. Rev. Fluid Mech. 39, 447–468.CrossRefGoogle Scholar
Faisst, H. & Eckhardt, B. 2003 Travelling waves in pipe flow. Phys. Rev. Lett. 91, 224–502.CrossRefGoogle ScholarPubMed
Fransson, J. & Alfredsson, P. 2003 On the hydrodynamic stability of channel flow with crossflow. Phys. Fluids 15, 436–441.Google Scholar
Goharzadeh, A., Khalili, A. & Jrgensen, B. B. 2005 Transition layer thickness at a fluid–porous interface. Phys. Fluids 17, 057102.CrossRefGoogle Scholar
Hains, F. D. 1967 Stability of plane Couette–Poiseuille flow. Phys. Fluids 10, 2079–2080.CrossRefGoogle Scholar
Hains, F. D. 1971 Stability of plane Couette–Poiseuille flow with uniform crossflow. Phys. Fluids 14, 1620–1623.CrossRefGoogle Scholar
Hamilton, J. M., Kim, J. & Waleffe, F. 1995 Regeneration mechanism of near-wall turbulence structures. J. Fluid Mech. 287, 317–348.Google Scholar
Hof, B., vanDoorne, C. W. H., Westerweel, J. & Nieuwstadt, F. T. M. 2005 Turbulence regeneration in pipe flow at moderate Reynolds numbers. Phys. Rev. Lett. 95, 214502.CrossRefGoogle ScholarPubMed
Hof, B., vanDoorne, 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 travelling waves in turbulent pipe flow. Science 305, 1594–1598.CrossRefGoogle ScholarPubMed
Joseph, D. D. 1968 Eigenvalue bounds for the Orr–Sommerfeld equation. J. Fluid Mech. 33, 617–621.Google Scholar
Joseph, D. D. 1969 Eigenvalue bounds for the Orr–Sommerfeld equation. Part 2. J. Fluid Mech. 36, 721–734.Google Scholar
Joslin, R. D. 1998 Aircraft laminar flow control. Annu. Rev. Fluid Mech. 30, 1–29.CrossRefGoogle Scholar
Kerswell, R. R. & Tutty, O. R. 2007 Recurrence of travelling waves in transitional pipe flow. J. Fluid Mech. 584, 69–102.CrossRefGoogle Scholar
Mack, L. M. 1976 A numerical study of the temporal eigenvalue spectrum of the Blasius boundary layer. J. Fluid Mech. 73, 497–520.Google Scholar
Majdalani, J., Zhou, C. & Dawson, C. A. 2002 Two-dimensional viscous flow between slowly expanding or contracting walls with weak permeability. J. Biomech. 35, 1399–1403.CrossRefGoogle ScholarPubMed
Mott, J. E. & Joseph, D. D. 1968 Stability of parallel flow between concentric cylinders. Phys. Fluids 11, 2065–2073.Google Scholar
Nicoud, F. & Angilella, J. R. 1997 Effects of uniform injection at the wall on the stability of Couette-like flows. Phys. Rev. E 56, 3000–3009.Google Scholar
Orszag, S. A. 1971 Accurate solution of the Orr–Sommerfeld stability equation. J. Fluid Mech. 50, 689–703.CrossRefGoogle Scholar
Peixinho, J. & Mullin, T. 2006 Decay of turbulence in pipe flow. Phys. Rev. Lett. 96, 094501.Google Scholar
Pfenniger, W. 1961 Transition in the inlet length of tubes at high Reynolds numbers. In Boundary Layer and Flow Control (ed. Lachman, G. V.), pp. 970–980. Pergamon.Google Scholar
Potter, M. C. 1966 Stability of plane Couette–Poiseuille flow. J. Fluid Mech 24, 609–619.Google Scholar
Reddy, S. C., Schmid, P. J. & Henningson, D. S. 1993 Pseudospectra of the Orr–Sommerfeld operator. SIAM J. Appl. Maths 53, 15–47.Google Scholar
Reynolds, W. C. & Potter, M. C. 1967 Finite-amplitude instability of parallel shear flows. J. Fluid Mech. 27, 465–492.CrossRefGoogle Scholar
Romanov, V. A. 1973 Stability of plane parallel Couette flow. Funct. Anal. Appl. 7, 137–146.CrossRefGoogle Scholar
Sadeghi, V. M. & Higgins, B. G. 1991 Stability of sliding Couette–Poiseuille flow in an annulus subject to axisymmetric and asymmetric disturbances. Phys. Fluids A 3, 2092–2104.Google Scholar
Sandblom, R. M. 2001 Filtering process. US Patent 4105547.Google Scholar
Schmid, P. J. 2007 Nonmodal stability theory. Annu. Rev. Fluid Mech. 39, 129–162.Google Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.CrossRefGoogle Scholar
Sheppard, D. M. 1972 Hydrodynamic stability of the flow between parallel porous walls. Phys. Fluids 15, 241–244.CrossRefGoogle Scholar
Squire, H. B. 1933 On the stability of three-dimensional disturbances of viscous flow between parallel walls. Proc. R. Soc. A 142, 621–628.Google Scholar
Vadi, P. K. & Rizvi, S. S. H. 2001 Experimental evaluation of a uniform transmembrane pressure crossflow microfiltration unit for the concentration of micellar casein from skim milk. J. Membr. Sci. 189, 69–82.CrossRefGoogle Scholar
Waleffe, F. 1997 On a self-sustaining mechanism in shear flows. Phys. Fluids 9, 883–900.Google Scholar
Wedin, H. & Kerswell, R. R. 2004 Exact coherent solutions in pipe flow: travelling wave solutions. J. Fluid Mech. 508, 333–371.CrossRefGoogle Scholar