Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-19T01:29:42.024Z Has data issue: false hasContentIssue false
  • English
  • Français

Steady, supercritical flow in collapsible tubes. Part 2. Theoretical studies

Published online by Cambridge University Press:  20 April 2006

Michael E. Mcclurken ,
Ifiyenia Kececioglu ,
Roger D. Kamm  and
Ascher H. Shapiro
Michael E. Mcclurken
Affiliation:
Fluid Mechanics Laboratory, Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, U.S.A.
Ifiyenia Kececioglu
Affiliation:
Fluid Mechanics Laboratory, Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, U.S.A.
Roger D. Kamm
Affiliation:
Fluid Mechanics Laboratory, Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, U.S.A.
Ascher H. Shapiro
Affiliation:
Fluid Mechanics Laboratory, Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Abstract

Theories are developed to explain the experimental observations of steady, supercritical flow in compliant, partially collapsed tubes, presented in the companion paper (part 1).

It is shown that the measured curves of area vs. distance are governed by a combination of (i) friction and gravity, which produce mean gradients of area, and (ii) longitudinal bending and tension forces, which produce standing waves of area superposed upon the mean gradients. The experiments confirm the one-dimensional theory for the mean gradients: (i) in the absence of gravity, friction causes a pressure rise and a positive mean gradient of area; (ii) a downward slope can cancel gravity and lead asymptotically to a uniform state having zero gradients of pressure and area.

The inviscid dispersion relationship for area waves due to longitudinal bending and tension is developed, based on a simple, approximate model for the mechanics of the tube. The phase velocity increases as the wavelength decreases, hence the group velocity exceeds the phase velocity. Consequently, in steady flows that are supercritical with respect to the infinite-wavelength phase velocity, energy can propagate upstream and standing waves of area may appear.

In the experiments of part 1, longitudinal tension predominated over longitudinal bending. The measured wavelengths of standing waves were found to be in general agreement with the dispersion relationship for tension-induced area waves. The observed streamwise growth of standing area waves is interpreted physically as the attenuation of waves radiating upstream from a source of disturbance such as a shock-like rapid increase of area. The rate of wave attenuation indicates that the skin-friction coefficient has a large out-of-phase oscillatory component. The observed steepness of shock transitions agrees with an approximate theory based on treating the forward portion of the shock as the rearward part of the standing wave train that the shock drives upstream.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 109 , August 1981 , pp. 391 - 415
Copyright
© 1981 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

Benjamin, K. B. 1959 Shearing flow over a wavy boundary. J. Fluid Mech. 6, 161–205.Google Scholar
Brower, R. W. 1970 Pressure—flow characteristics of collapsible tubes. Ph.D. thesis, University of Pennsylvania.
Flaherty, J. E., Keller, J. B. & Rubinow, S. I. 1972 Post-buckling behaviour of elastic tubes and rings with opposite sides in contact. SIAM J. Appl. Math. 23, 446–455.Google Scholar
Hornbeck, R. W. 1975 Numerical Methods, p. 195. New York: Quantum.
Hsu, S. 1968 Turbulent flow in wavy pipes. Ph.D. thesis, University of Iowa.
Hsu, S. & Kennedy, J. F. 1971 Turbulent flow in wavy pipes. J. Fluid Mech. 47, 481–502.Google Scholar
Jan, D. L. 1980 Refilling of collapsible tubes. M.S. thesis, Massachusetts Institute of Technology.
Kececioglu, I. 1979 Experimental determination of the structure of shock waves in fluid flow through collapsible tubes with application to the design of a flow regulator. Thesis for the Degree of Mechanical Engineer, Massachusetts Institute of Technology.
Kececioglu, I., McClurken, M. E., Kamm, R. D. & Shapiro, A. H. 1981 Steady, supercritical flow in collapsible tubes. Part 1. Experimental observations. J. Fluid Mech. 109, 367–389.Google Scholar
Mcclurken, M. E. 1980 Supercritical flow in collapsible tubes. Ph.D. thesis, Department of Mechanical Engineering, Massachusetts Institute of Technology.
Schlichting, H. 1968 Boundary Layer Theory, 6th edn. McGraw-Hill.
Shapiro, A. H. 1977 Steady flow in collapsible tubes. Trans. A.S.M.E., J. Biomech. Engng 99, 126–147.Google Scholar
Sorrell, F. Y. & Sturm, G. V. 1977 The interaction between capillary waves and a current and wave decay. J. Fluid Mech. 81, 241–256.Google Scholar
Swidler, S. C. 1980 Effect of longitudinal tension and curvature on pressure—area relationships of collapsible tubes. M.S. thesis, Dept. of Mechanical Eng., Massachusetts Institute of Technology.
Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley.
Zilker, D. P., Cook, G. W. & Hanratty, T. J. 1977 Influence of the amplitude of a solid wavy wall on a turbulent flow. Part 1. Non-separated flows. J. Fluid Mech. 82, 29–51.Google Scholar