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Steady and unsteady separation in an approximately two-dimensional indented channel

Published online by Cambridge University Press:  20 April 2006

C. D. Bertram  and
T. J. Pedley
C. D. Bertram
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW Present address: Centre for Biomedical Engineering, University of New South Wales, P.O. Box 1, Kensington, Australia 2033.
T. J. Pedley
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW
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Abstract

Experiments are performed on steady and impulsively started flow in an approximately two-dimensional closed channel, with one wall locally indented. In plan the indentation is a long trapezium which halves the channel width: the inclination of the sloping walls is approximately 5.7°, and these tapered sections merge smoothly into the narrowest section via rounded corners. The Reynolds number $ Re = a_0\overline{u}_0/\nu $ (a0 = unindented channel width, $\overline{u}_0$ = steady mean velocity in the unindented channel) lies in the range 300 [les ] Re [les ] 1800. In steady flow, flow visualization reveals that separation occurs on the lee slope of the indentation, at a distance downstream of the convex corner which decreases (tending to a non-zero value) as Re increases. There is no upstream separation, and there is some evidence of three-dimensionality of the flow in the downstream separated eddy. Pressure measurements agree qualitatively but not quantitatively with theoretical predictions. Unsteady flow visualization reveals that, as in external flow, wall-shear reversal occurs over much of the lee slope (at dimensionless time $\tau = \overline{u}_0t/a_0 \approx 4$) before there is any evidence of severe boundary-layer thickening and breakaway. Then, at τ ≈ 5.5, a separated eddy develops, and its nose moves gradually upstream from the downstream end of the indentation to its eventual (τ ≈ 75) steady-state position on the lee slope. At about the same time as the wall-shear reversal, wavy vortices appear at the edge of the boundary layer on both walls of the channel, and (for Re < 750) subsequently disappear again; these are interpreted as manifestations of inflection-point instability and not as intrinsic aspects of boundary-layer separation. Pressure measurements are made to investigate the discrepancy between the actual pressure drop across the lee slope and that predicted on the assumption that energy dissipation is quasi-steady. This discrepancy has a maximum value of approximately $1.5\rho \overline{u}^2_0$ (ρ = fluid density), and decays to zero by the time τ ≈ 7.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 130 , May 1983 , pp. 315 - 345
Copyright
© 1983 Cambridge University Press

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References

Åström, K. J. & Eykhoff, P. 1971 System identification – a survey Automatica 7, 123–162.Google Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Bertram, C. D. 1982 Two modes of instability in a thick-walled collapsible tube conveying a flow J. Biomech. 15, 223–224.Google Scholar
Bertram, C. D. & Pedley, T. J. 1982 A mathematical model of unsteady collapsible tube behaviour J. Biomech. 15, 39–50.Google Scholar
Bonis, M. 1979 Écoulement visqueux permanent dans un tube collabable elliptique. Thèse de Doctorat d’État, Université de Technologie de Compiègne.
Bouard, R. & Coutanceau, M. 1980 The early stage of development of the wake behind an impulsively started cylinder for 40 < Re < 104. J. Fluid Mech. 101, 583–607.
Cebeci, T. 1979 The laminar boundary layer on a circular cylinder started impulsively from rest J. Comp. Phys. 31, 153–172.Google Scholar
Collins, W. M. & Dennis, S. C. R. 1973a The initial flow past an impulsively started circular cylinder Q. J. Mech. Appl. Maths 26, 53–75.Google Scholar
Collins, W. M. & Dennis, S. C. R. 1973b Flow past an impulsively started circular cylinder J. Fluid Mech. 60, 105–127.Google Scholar
Conrad, W. A. 1969 Pressure-flow relationships in collapsible tubes IEEE Trans. Biomed. Engng 16, 284–295.Google Scholar
Cowley, S. J. 1983 Steady flow through distorted rectangular tubes of large aspect ratio Proc. R. Soc. Lond. A385, 107–127.Google Scholar
Koromilas, C. A. & Telionis, D. P. 1980 Unsteady laminar separation: an experimental study. J. Fluid Mech. 97, 347–384.
Mcalister, K. W. & Carr, L. W. 1979 Water tunnel visualisations of dynamic stall. Trans. ASME I: J. Fluids Engng 101, 376–380.
Pedley, T. J. 1980 The Fluid Mechanics of Large Blood Vessels. Cambridge University Press.
Proudman, I. & Johnson, K. 1962 Boundary layer growth near a rear stagnation point J. Fluid Mech. 14, 161–168.Google Scholar
Schlichting, H. 1968 Boundary Layer Theory, 6th edn. McGraw-Hill.
Sears, W. R. & Telionis, D. P. 1975 Boundary-layer separation in unsteady flow SIAM J. Appl. Maths 28, 215–235.Google Scholar
Secomb, T. W. 1979 Flows in tubes and channels with indented and moving walls. Ph.D. thesis, Cambridge University.
Smith, F. T. 1974 Boundary layer flow near a discontinuity in wall conditions J. Inst. Maths Applics 13, 127–145.Google Scholar
Smith, F. T. 1976a Flow through constricted or dilated pipes and channels: Part 1. Q. J. Mech. Appl. Maths 29, 343–364.
Smith, F. T. 1976b Flow through constricted or dilated pipes and channels: Part 2. Q. J. Mech. Appl. Maths 29, 365–376.
Smith, F. T. 1977 Upstream interactions in channel flows J. Fluid Mech. 79, 631–655.Google Scholar
Smith, F. T. & Daniels, P. G. 1981 Removal of Goldstein's singularity at separation, in flow past obstacles in wall layers J. Fluid Mech. 110, 1–37.Google Scholar
Smith, F. T. & Duck, P. W. 1980 On the severe non-symmetric constriction, curving or cornering of channel flows J. Fluid Mech. 98, 727–753.Google Scholar
UR. A. & GORDON, M. 1970 Origin of Korotkoff sounds Am. J. Physiol. 218, 524–529.Google Scholar
Walker, J. D. A. 1978 The boundary layer due to rectilinear vortex Proc. R. Soc. Lond. A359, 167–188.Google Scholar
Young, D. F. & Tsai, F. Y. 1973a Flow characteristics in models of arterial stenosis – I. Steady flow J. Biomech. 6, 395–410.Google Scholar
Young, D. F. & Tsai, F. Y. 1973b Flow characteristics in models of arterial stenosis – II. Unsteady flow J. Biomech. 6, 547–559.Google Scholar