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Closed-cavity laminar flows at moderate Reynolds numbers

Published online by Cambridge University Press:  20 April 2006

J. F. Brady  and
A. Acrivos
J. F. Brady
Affiliation:
Department of Chemical Engineering, Stanford University, Stanford, CA 94305 Present address: Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139.
A. Acrivos
Affiliation:
Department of Chemical Engineering, Stanford University, Stanford, CA 94305
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Abstract

The paradox reported by Brady & Acrivos (1981) of the non-existence of similarity solutions in the Reynolds-number range 10·25 < R < 147 for the flow in a tube with an accelerated surface velocity is resolved. It is shown that the source of the difficulty lies in the assumption that the tube is infinite in extent. For a finite tube, it is demonstrated that the presence of the closed end, even though far removed from the origin, affects in a fundamental way the structure of the flow throughout the entire tube. The change in the flow structure that occurs in a finite tube at R = 10·25 is caused by the fluid which is returning from the downstream end; it is shown further that the problem of determining the motion in a long finite tube is equivalent to that of selecting the initial condition for the boundary-layer equations that properly takes into account the presence of the reverse flow. By applying a method originally developed by Klemp & Acrivos (1976) for selecting this condition, the flow in a finite tube is determined numerically for Reynolds numbers up to 70. In addition, it is shown that the same change in structure brought about by the returning fluid occurs in a finite two-dimensional channel at R = 57, even though the corresponding similarity solutions exist for all values of R. The results suggest that similarity solutions should be viewed with caution because they may not represent a real flow once a critical Reynolds number is exceeded.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 115 , February 1982 , pp. 427 - 442
Copyright
© 1982 Cambridge University Press

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References

Barenblatt, G. I. & Zel'dovich, Ya. B. 1972 Self-similar solutions as intermediate asymptotics. Ann. Rev. Fluid Mech. 4, 285–312.Google Scholar
Brady, J. F. & Acrivos, A. 1981 Steady flow in a channel or tube with an accelerating surface velocity. An exact solution to the Navier — Stokes equations with reverse flow. J. Fluid Mech. 112, 127–150.Google Scholar
Brady, J. F. & Acrivos, A. 1982 The deformation and breakup of a slender drop in extensional flow: inertial effects. J. Fluid Mech. 115, 443–451.Google Scholar
Klemp, J. B. & Acrivos, A. 1972 A method for integrating the boundary-layer equations through a region of reverse flow. J. Fluid Mech. 53, 177–191.Google Scholar
Klemp, J. B. & Acrivos, A. 1976 A moving-wall boundary layer with reverse flow. J. Fluid Mech. 76, 363–381.Google Scholar
Moffatt, H. K. & Duffy, B. R. 1980 Local similarity solutions and their limitations. J. Fluid Mech. 96, 299–313.Google Scholar
Roache, P. J. 1972 Computational Fluid Dynamics, p. 73. Hermosa.
Van Dyke, M. 1970 Entry flow in a channel. J. Fluid Mech. 44, 813–823.Google Scholar