Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-19T00:26:50.988Z Has data issue: false hasContentIssue false
  • English
  • Français

Upstream interactions in channel flows

Published online by Cambridge University Press:  11 April 2006

F. T. Smith
F. T. Smith
Affiliation:
Department of Mathematics, Imperial College, London
Get access Rights & Permissions [Opens in a new window]

Abstract

The manner in which fluid driven through a channel of width a responds in anticipation of a severe asymmetric distortion (e.g. to the wall or interior conditions downstream) is discussed when the oncoming flow is fully developed, the characteristic Reynolds number K is large and the whole motion remains laminar. Far ahead of the disturbance, at distances $O(aK^{\frac{1}{7}})$, there occurs a free interaction which triggers a small displacement in the core, generating viscous layers near both walls; the relatively large induced pressure gradient acting across the channel is then found to sustain the growth of this displacement. Numerical solutions of the fundamental nonlinear problem show that one layer separates in a regular fashion, but that beyond separation the other layer, under compression, produces a singularity in the interaction. Analysis of the singularity based on a self-similar structure in the partly reversed flow then leads to a description of the nonlinear flow features nearer the distortion which seems to have strong physical significance. The major implication is that the flow will have already separated at one wall, and developed a definite nonlinear character quite distinct from that of the original Poiseuille flow, long before it reaches finite distances from the distortion. The separation point is predicted to be a distance $0.49 aK^{\frac{1}{7}} (+ O(a))$ ahead of the particular finite distortion. Comparisons of this and other predictions with computed solutions of the full Navier–Stokes equations show reasonable agreement.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 79 , Issue 4 , 23 March 1977 , pp. 631 - 655
Copyright
© 1977 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

Friedman, M. 1972 Laminar flow in a channel with a step. J. Engng Math. 6, 285.Google Scholar
Goldstein, S. 1948 On laminar boundary layer flow near a point of separation. Quart. J. Mech. Appl. Math. 1, 43.Google Scholar
Greenspan, D. 1969 Numerical studies of steady, viscous, incompressible flow in a channel with a step. J. Engng Math. 3, 21.Google Scholar
Hurd, A. C. & Peters, A. R. 1970 Analysis of flow separation in a confined two-dimensional channel. Trans. A.S.M.E., J. Basic Engng, 92, 908.Google Scholar
Keller, H. B. & Cebeci, T. 1971 Accurate numerical methods for boundary layer flows. I. Two-dimensional laminar flows. 2nd Int. Conf. Num. Meth. in Fluid Dyn., p. 92. Springer.
Matida, Y., Kuwahara, K. & Takami, H. 1975 Numerical study of steady two-dimensional flow past a square cylinder in a channel. J. Phys. Soc. Japan, 38, 1522.Google Scholar
Messiter, A. F. 1970 Boundary layer flow near the trailing edge of a flat plate. S.I.A.M. J. Appl. Math. 18, 241.Google Scholar
Reyhner, T. A. & Flügge-Lotz, I. 1968 The interaction of a shock wave with a laminar boundary layer. Int. J. Nonlinear Mech. 3, 173.Google Scholar
Smith, F. T. 1973 Laminar flow over a small hump on a flat plate. J. Fluid Mech. 57, 803.Google Scholar
Smith, F. T. 1974 Boundary layer flow near a discontinuity in wall conditions. J. Inst. Math. Appl. 13, 127.Google Scholar
Smith, F. T. 1976a Flow through constricted or dilated pipes and channels. Quart. J. Mech. Appl. Math. 29, 343.Google Scholar
Smith, F. T. 1976b Flow through constricted or dilated pipes and channels. Part 2. Quart. J. Mech. Appl. Math. 29, 365.Google Scholar
Smith, F. T. 1976c On entry-flow effects in bifurcating, blocked or constricted tubes. J. Fluid Mech. 78, 709.Google Scholar
Smith, F. T. & Stewartson, K. 1973 On slot-injection into a supersonic laminar boundary layer. Proc. Roy. Soc. A 332, 1.Google Scholar
Stewartson, K. 1969 On the flow near the trailing edge of a flat plate II. Mathematika, 16, 106.Google Scholar
Stewartson, K. 1970 On supersonic laminar boundary layers near convex corners. Proc. Roy. Soc. A 319, 289.Google Scholar
Stewartson, K. 1974 Multistructured boundary layers on flat plates and related bodies. Adv. in Appl. Mech. 14, 145.Google Scholar
Stewartson, K. & Williams, P. G. 1969 Self-induced separation. Proc. Roy. Soc. A 312, 181.Google Scholar
Stewartson, K. & Williams, P. G. 1973 Self-induced separation II. Mathematika, 20, 98.Google Scholar
Wilson, S. D. R. 1969 The development of Poiseuille flow. J. Fluid Mech. 38, 793.Google Scholar