Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-18T18:38:32.861Z Has data issue: false hasContentIssue false
  • English
  • Français

Stability bounds on turbulent Poiseuille flow

Published online by Cambridge University Press:  21 April 2006

G. R. Ierley  and
W. V. R. Malkus
G. R. Ierley
Affiliation:
Fluids Research Oriented Group, Department of Mathematical Sciences, Michigan Technological University, Houghton, MI 49931, USA
W. V. R. Malkus
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

For steady-state turbulent flows with unique mean properties, we determine a sense in which the mean velocity is linearly supercritical. The shear-turbulence literature on this point is ambiguous. As an example, we reassess the stability of mean profiles in turbulent Poiseuille flow. The Reynolds & Tiederman (1967) numerical study is used as a starting point. They had constructed a class of one-dimensional flows which included, within experimental error, the observed profile. Their numerical solutions of the resulting Orr-Sommerfeld problems led them to conclude that the Reynolds number for neutral infinitesimal disturbances was twenty-five times the Reynolds number characterizing the observed mean flow. They found also that the first nonlinear corrections were stabilizing. In the realized flow, this latter conclusion appears incompatible with the former. Hence, we have sought a more complete set of velocity profiles which could exhibit linear instability, retaining the requirement that the observed velocity profile is included in the set. We have added two dynamically generated modifications of the mean. The first addition is a fluctuation in the curvature of the mean flow generated by a Reynolds stress whose form is determined by the neutrally stable Orr-Sommerfeld solution. We find that this can reduce the stability of the observed flow by as much as a factor of two. The second addition is the zero-average downstream wave associated with the above Reynolds stress. The three-dimensional linear instability of this modification can even render the observed flow unstable. Those wave amplitudes that just barely will ensure instability of the observed flow are determined. The relation of these particular amplitudes to the limiting conditions admitted by an absolute stability criterion for disturbances on the mean flow is found. These quantitative results from stability theory lie in the observationally determined Reynolds-Tiederman similarity scheme, and hence are insensitive to changes in Reynolds number.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 187 , February 1988 , pp. 435 - 449
Copyright
© 1988 Cambridge University Press

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

Busse, F. H. 1969. Z. Agnew. Math. Phys. 20, 1.
Busse, F. H. 1978 Adv. Appl. Mech. 18, 77.
Chan, S. K. 1971 Stud. Appl. Maths 50, 13.
Claussen, M 1983 Boundary-Layer Met. 27, 209.
Herbert, T. 1980 AIAA J. 18, 243.
Herring, J. 1963 J. Atmos. Sci. 20, 325.
Herring, J. 1964 J. Atmos. Sci. 21, 277.
Joseph, D. D. 1976 Stability of Fluid Motions. Springer.
Joseph, D. D. & Carmi, S. 1969 Q. Appl. Maths. 26, 575.
Kraichnan, R. H. 1962 Phys. Fluids 5, 1374.
Malkus, W. V. R. 1956 J. Fluid Mech. 1, 521.
Malkus, W. V. R. 1979 J. Fluid. Mech. 90, 401.
Malkus, W. V. R. & Proctor, M. R. E. 1975 J. Fluid Mech. 67, 417.
Orszag, S. A. 1971 J. Fluid Mech. 50, 689.
Orszag, S. A. & Patera, A. T. 1981 J. Fluid Mech. 128, 347.
Reynolds, W. C. & Tiederman, W. G. 1967 J. Fluid Mech. 27, 253.
Zaff, D. 1987 Secondary instability in Ekman boundary flow. Ph.D. Thesis, Department of Mathematics, Massachusetts Institute of Technology.