Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-29T13:59:10.907Z Has data issue: false hasContentIssue false
  • English
  • Français

A nonlinear instability burst in plane parallel flow

Published online by Cambridge University Press:  29 March 2006

L. M. Hocking ,
K. Stewartson ,
J. T. Stuart  and
S. N. Brown
L. M. Hocking
Affiliation:
Department of Mathematics, University College, London
K. Stewartson
Affiliation:
Department of Mathematics, University College, London
J. T. Stuart
Affiliation:
Department of Mathematics, Imperial College, London
S. N. Brown
Affiliation:
University College, London
Get access Rights & Permissions [Opens in a new window]

Abstract

An infinitesimal centre disturbance is imposed on a fully Ldveloped plane Poiseuille flow at a Reynolds number R slightly greater than the critical value Rc for instability. After a long time, t, the disturbance consists of a modulated wave whose amplitude A is a slowly varying function of position and time. In an earlier paper (Stewartson & Stuart 1971) the parabolic differential equation satisfied by A for two-dimensional disturbances was found; the theory is here extended to three dimensions. Although the coefficients of the equation are coinples, a start is made on elucidating the properties of its solutions by assuming that these coefficients are real. It is then found numerically and confirmed analytically that, for a finite value of (R-Rc)t, the amplitude A develops an infinite peak at the wave centre. The possible relevance of this work to the phenomenon of transition is discussed.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 51 , Issue 4 , 22 February 1972 , pp. 705 - 735
Copyright
© 1972 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

Anderson, D. L. N. & Derrick, G. H. 1970 J. Math. Phys. 11, 1336–1346.
Benjamin, T. B. 1961 J. Fluid Mech. 10, 401–419.
Benney, D. J. 1961 J. Fluid Mech. 10, 209–236.
Benney, D. J. 1964 Phys. Fluids, 7, 319–326.
Benney, D. J. & Lin, C. C. 1960 Phys. Fluids, 3, 656–657.
Craik, A. D. D. 1971 J. Fluid Mech. 50, 393–413.
Criminale, W. O. & Kovasznay, L. S. G. 1962 J. Fluid Mech. 14, 59–80.
Davies, S. J. & White, C. M. 1928 Proc. Roy. Soc. A 119, 92–107.
DiPrima, R., Ecphaus, W. & Segel, L. A. 1971 J. Fluid Mech. 49, 705–744.Google Scholar
Gaster, M. 1968a Phys. Fluids, 11, 723–727.
Gaster, M. 1968b J. Fluid Mech. 32, 173–184.
Greenspan, H. P. & Benney, D. J. 1963 J. Fluid Mech. 15, 133–153.
Hocking, L. M. & Stewartson, K. 1971 Mathernatika, 18, 219–239.
Hocking, L. M. & Stewartson, K. 1972 Proc. Roy. Soc. A 326, 289–313.
Klebanoff, P. S., Tidstrom, K. D. & Sargent, L. M. 1962 J. Fluid Mech. 12, 1–34.Google Scholar
Komoda, H. 1967 Phys. Fluids, 10 (suppl.), 87–94.
Kovasznay, L. S. G., Komoda, H. & Vasudeva, B. R. 1962 Proc. Heat Transfer and Fluid Mechanics Institute, pp. 1–26. Stanford University Press.
Lin, C. C. & Benney, D. J. 1966 Proc. XIth. Int. Cong. AppZ. Mech., Munich, pp. 797–802, Springer.
Michael, D. H. 1961 J. Fluid Mech. 10, 525–528.
Mollo-Christensen, E. 1971 A.I.A.A.J. 9, 1217–1220.
Newell, A. C. & Whitehead, J. A. 1969 J. Fluid Mech. 38, 279–304.
Peperis, C. L. & Shkoller, B. 1967 J. Fluid Mech. 29, 31–38.
Pekeris, C. L. & Sheoller, B. 1969 J. Fluid Mech. 39, 629–639.
Raetz, G. S. 1959 Northrop Rep. NOR–59–383 (BLC–121).
Reynolds, W. C. & Potter, M. C. 1967 J. Fluid Mech. 27, 465–492.Google Scholar
Segel, L. A. 1969 J. Fluid Mech. 38, 203–224.
Squire, H. B. 1933 Proc. Roy. Soc. A 142, 621–628.
Stewartson, K. & Stuart, J. T. 1971 J. Fluid Mech. 48, 529–545.
Stuart, J. T. 1958 J. Fluid Mech. 4, 1–22.
Stuart, J. T. 1960 J. Fluid Mech. 9, 353–370.
Stuart, J. T. 1961 Adv. in Aero. Sci. 3, 121–142.
Stuart, J. T. 1962 Proc. Int. Cong. Appl. Mech., Stresa, pp. 63–97. Elsevier.
Stuart, J. T. 1965 NPL Aero. Rep. no. 1147. (See also Agard Rep. no. 514.)
Tani, I. 1969 Annual Review of Fluid Mechanics, I (ed. W. R. Sears), pp. 169–196. Annual Reviews Inc.
Van Dyke, M. D. 1964 Perturbation Methods in Pluid Mechanics, p. 2. Academic Press.
Watanabe, T. 1969 J. Phys. Soc. Japan, 27, 1341–1350.
Watson, J. 1960a J. Fluid Mech. 9, 371–389.
Watson, J. 1960b Proc. Roy. Soc. A 254, 562–569.