Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-18T15:41:19.989Z Has data issue: false hasContentIssue false
  • English
  • Français

Stability of the developing laminar flow in a parallel-plate channel

Published online by Cambridge University Press:  28 March 2006

T. S. Chen  and
E. M. Sparrow
T. S. Chen
Affiliation:
University of Minnesota, Minneapolis, Minnesota Present address: Department of Mechanical Engineering, University of Missouri at Rolla, Rolla, Missouri.
E. M. Sparrow
Affiliation:
University of Minnesota, Minneapolis, Minnesota
Get access Rights & Permissions [Opens in a new window]

Abstract

The hydrodynamic stability of the developing laminar flow in the entrance region of a parallel-plate channel is investigated using the theory of small disturbances. The stability of the fully developed flow is also re-examined. A wide range of analytical (i.e. asymptotic) and numerical methods are employed in the stability investigation. Among the asymptotic methods, each of three viscous solutions (singular, regular and composite) is used along with the inviscid solution to provide critical Reynolds numbers and complete neutral stability curves. Two numerical methods, finite differences and stepwise integration, are applied to calculate critical Reynolds numbers. The basic flow in the development region is treated from two stand-points: as a channel velocity profile and as a boundary-layer velocity profile. Extensive comparisons among the various methods and flow models disclose their various strengths and ranges of applicability. As a general result, it is found that the critical Reynolds number decreases monotonically with increasing distance from the channel entrance, approaching the fully developed value as a limit.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 30 , Issue 2 , 9 November 1967 , pp. 209 - 224
Copyright
© 1967 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

Chen, T. S. 1966 Hydrodynamic stability of developing flow in a parallel-plate channel. Ph.D. Thesis, Department of Mechanical Engineering, University of Minnesota.
Chen, T. S., Joseph, D. D. & Sparrow, E. M. 1966 Phys. Fluids, 9, 2519.
Fu, T. S. 1967 Viscous instability of asymmetric parallel flows in channels. Ph.D. Thesis Department of Aeronautics and Engineering Mechanics, University of Minnesota.
Gröhne, D. 1954 Z. Angew. Math. Mech. 34, 344.
Hahneman, E., Freeman, J. C. & Finston, M. 1948 J. Aero. Sci. 15, 493.
Heisenberg, W. 1924 Ann. Phys. 74, 577.
Holstein, H. 1950 Z. Angew. Math. Mech. 30, 25.
Kurtz, E. F. & Crandall, S. H. 1962 J. Math. Phys. 41, 264.
Lin, C. C. 1945 Quart. Appl. Math. 3, 117, 213, 277.
Lin, C. C. 1955 The Theory of Hydrodynamic Stability. Cambridge University Press.
Lock, R. C. 1954 Proc. Camb. Phil. Soc. 50, 105.
Miles, J. W. 1960 J. Fluid Mech. 8, 593.
Nachtsheim, P. R. 1964 NASA TN D-2414.
Reid, W. H. 1965 The stability of parallel flows. In Basic Development in Fluid Mechanics. Ed. by M. Holt. New York: Academic Press.
Schlichting, H. 1934 Z. Angew. Math. Mech. 14, 368.
Schlichting, H. 1960 Boundary Layer Theory. New York: McGraw-Hill.
Shen, S. F. 1964 Stability of laminar flows. In Theory of Laminar Flows. Ed. by F. K. Moore, Princeton University Press.
Sparrow, E. M., Lin, S. H. & Lundgren, T. S. 1964 Phys. Fluids, 7, 338.
Stuart, J. T. 1963 Hydrodynamic stability. In Laminar Boundary Layers. Ed. by L. Rosenhead. Oxford University Press.
Thomas, L. H. 1953 Phys. Rev. 91, 780.
Tietjens, O. 1925 Z. Angew. Math. Mech. 5, 200.
Tollmien, W. 1929 Nachr. Ges. Wiss. Göttingen, Math.-Phys. Klasse, p. 21.
Tollmien, W. 1947 Z. Angew. Math. Mech. 25–27, 33, 70.
Tsou, F. K. 1965 Velocity field, hydrodynamic stability, and heat transfer for boundary-layer flow along a continuous moving surface. Ph.D. Thesis, Department of Mechanical Engineering, University of Minnesota.