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The laminar decay of suddenly blocked channel and pipe flows

Published online by Cambridge University Press:  29 March 2006

S. Weinbaum  and
K. H. Parker
S. Weinbaum
Affiliation:
Physiological Flow Studies Unit, Department of Aeronautics, Imperial College, London Present address: Department of Mechanical Engineering, The City College of The City of New York, New York 10031.
K. H. Parker
Affiliation:
Physiological Flow Studies Unit, Department of Aeronautics, Imperial College, London
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Abstract

This paper is a theoretical investigation of the stable laminar decay of a fully established channel or pipe flow following a sudden blockage such as would be caused by the rapid closure of a valve or imposition of an end wall or gate. The development of the subsequent velocity and pressure fields is examined from the instant the initial pressure wave passes until the final decay of all motion. Three time scales of hydrodynamic interest are identified and the relevant solutions are obtained. The time scales are as follows: (i) a very short time characteristic of the passage of the pressure wave during which the velocity field adjusts inviscidly to the new boundary conditions imposed by the presence of the end wall, (ii) a short diffusion time during which the displacement interaction generated by the diffusion of the primary Rayleigh layer induces a substantial secondary motion with distinct side-wall boundary layers and an inviscid core and (iii) a long diffusion time during which the boundary layers fill the entire channel or pipe and the residual motion then dies out. The secondary flow for short diffusion times is of special interest in that it is an example of an unsteady boundary layer where the external pressure gradient and inviscid outer flow are unknown and determined by the integrated time history of the combined mass flow displacement generated by the primary- and secondary-flow boundary layers. The paper closes with some preliminary comments and experimental observations on decelerating pipe flows.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 69 , Issue 4 , 24 June 1975 , pp. 729 - 752
Copyright
© 1975 Cambridge University Press

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References

Batchelor, G. K. 1967 An Introduction to Fluid Mechanics, $4.3. Cambridge University Press.
Cabslaw, H. S. & Jaeger, J. C. 1959 Conduction of Heat in Solids, 2nd edn, $12.5. Oxford University Press.
Faller, A. J. 1963 An experimental study of the instability of the laminar Ekman boundary layer J. Fluid Mech. 15, 560–576.Google Scholar
Goldstein, S. 1938 Modern Developments in Fluid Mechanics, vol. 1, chap. 7. Oxford University Press.
Goldstein, S. & Rosenhead, L. 1936 Boundary layer growth Proc. Camb. Phil. Soc. 32, 392–401.Google Scholar
Gregory, N., Stuart, J. T. & Walker, W. S. 1955 On the stability of three-dimensional boundary layers with application to the flow due to a rotating disk. Phil. Trans A 248, 155–199.Google Scholar
Kelly, R. E. 1962 The final approach to steady, viscous flow near a stagnation point following a change in free stream velocity J. Fluid Mech. 13, 449–464.Google Scholar
Lilly, D. K. 1966 On the instability of Ekman boundary flow J. Atmos. Sci. 23, 481–494.Google Scholar
Nerem, R. M. & Seed, W. A. 1972 In vivo study of the nature of aortic flow disturbances. Cardiovasc. Res. 6, 1–14.Google Scholar
Schlichting, H. 1934 Laminare Kanaleinlaufströmung Z. angew. Math. Mech. 14, 368–373.Google Scholar
Seed, W. A. & Wood, N. B. 1971 Velocity patterns in the aorta Cardiovasc. Res. 5, 319–330.Google Scholar
Stuart, J. T. 1963 In Laminar Boundary Layers (ed. L. Rosenhead), chap. 7, pp. 349–408. Oxford: Clarendon Press.
Stuart, J. T. 1971 Unsteady boundary layers. IUTAM Symp. on Unsteady Boundary layers (1971). Quebec: Laval University Press.
Szymanski, P. 1932 Quelques solutions exactes des équations de l'hydrodynamique du fluide visqueux dans le cas d'un tube cylindrique. J. Math. Pure Appl. 11 (9), 67.Google Scholar
Tokuda, N. 1970 Uniformly convergent series solution for unsteady stagnation flows. In Fluid Dyn. Trans., vol. 5, part 2, pp. 175–191. Warsaw: Polish Acad. Sci.
VAN DYKE, M. 1970 Entry flow in a channel J. Fluid Mech. 44, 813–823.Google Scholar
Wilson, S. D. R. 1971 Entry flow in a channel. Part 2 J. Fluid Mech. 46, 787–799.Google Scholar