Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-19T14:55:31.546Z Has data issue: false hasContentIssue false
  • English
  • Français

Flow in a channel with a moving indentation

Published online by Cambridge University Press:  21 April 2006

M. E. Ralph  and
T. J. Pedley
M. E. Ralph
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK
T. J. Pedley
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

The unsteady flow of a viscous, incompressible fluid in a channel with a moving indentation in one wall has been studied by numerical solution of the Navier-Stokes equations. The solution was obtained in stream-function-vorticity form using finite differences. Leapfrog time-differencing and the Dufort-Frankel substitution were used in the vorticity transport equation, and the Poisson equation for the stream function was solved by multigrid methods. In order to resolve the boundary-condition difficulties arising from the presence of the moving wall, a time-dependent transformation was applied, complicating the equations but ensuring that the computational domain remained a fixed rectangle.

Downstream of the moving indentation, the flow in the centre of the channel becomes wavy, and eddies are formed between the wave crests/troughs and the walls. Subsequently, certain of these eddies ‘double’, that is a second vortex centre appears upstream of the first. These observations are qualitatively similar to previous experimental findings (Stephanoff et al. 1983, and Pedley & Stephanoff 1985), and quantitative comparisons are also shown to be favourable. Plots of vorticity contours confirm that the wave generation process is essentially inviscid and reveal the vorticity dynamics of eddy doubling, in which viscous diffusion and advection are important at different stages. The maximum magnitude of wall vorticity is found to be much larger than in quasi-steady flow, with possibly important biomedical implications.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 190 , May 1988 , pp. 87 - 112
Copyright
© 1988 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

Armaly, B. F., Durst, F., Pereira, J. C. F. & Schonung, B. 1983 Experimental and theoretical investigation of backward-facing step flow. J. Fluid Mech. 127, 473–496.Google Scholar
Bertram, C. D. & Pedley, T. J. 1983 Steady and unsteady separation in an approximately two-dimensional indented channel. J. Fluid Mech. 130, 315–345.Google Scholar
Borgas, M. S. 1986 Waves, singularities and non-uniqueness in channel and pipe flows. Ph.D. dissertation, Cambridge University.
Brandt, A. 1977 Multi-level adaptive solutions to boundary-value problems. Maths Comp. 31, 333–390.Google Scholar
Cancelli, C. & Pedley, T. J. 1985 A separated-flow model for collapsible-tube oscillations. J. Fluid Mech. 157, 375–404.Google Scholar
Cherdron, W., Durst, F. & Whitelaw, J. H. 1978 Asymmetric flows and instabilities in symmetric ducts with sudden expansions. J. Fluid Mech. 84, 13–31.Google Scholar
Daly, B. J. 1974 A numerical study of pulsatile flow through constricted arteries. In Proc. 4th Intl Conf. on Numerical Methods in Fluid Dynamics. (ed. R. D. Richtmeyer). Lecture Notes in Physics, vol. 35, pp. 117–124. Springer.
Dritschel, D. G. 1988 Contour surgery: a topological reconnection scheme for extended integrations using contour dynamics. J. Comput. Phys (in the press).Google Scholar
Duck, P. W. 1985 Laminar flows over unsteady humps: the formation of waves. J. Fluid Mech. 160, 465–498.Google Scholar
Hung, T. -K. & Schuessler, G. B. 1977 An analysis of the hemodynamics of the opening of aortic valves. J. Biomech. 10, 597–606.Google Scholar
Ku, D. N., Giddens, D. P., Zarins, C. K. & Glagov, S. 1985 Pulsatile flow and atherosclerosis in the human carotid bifurcation: positive correlation between plaque location and low and oscillatory shear stress. Arteriosclerosis 5, 293–302.CrossRefGoogle Scholar
McKee S. & Mitchell, A. R. 1970 Alternating direction methods for parabolic equations in two space dimensions with a mixed derivative. Comp. J. 13, 81–86.Google Scholar
Pedley, T. J. & Stephanoff, K. D. 1985 Flow along a channel with a time-dependent indentation in one wall: the generation of vorticity waves. J. Fluid Mech. 160, 337–367.Google Scholar
Peskin, C. S. 1972 Flow patterns around heart valves: a numerical method. J. Comput. Phys. 10, 252–271.Google Scholar
Peskin, C. S. 1977 Numerical analysis of blood flow in the heart. J. Comput. Phys. 25, 220–252.Google Scholar
Ralph, M. E. 1986 Oscillatory flows in wavy-walled tubes. J. Fluid Mech. 168, 515–540.Google Scholar
Roache, P. J. 1976 Computational Fluid Dynamics, 2nd edn. Albuquerque: Hermosa.
Robertson, J. M., Clark, M. E. & Cheng, L. C. 1982 A study of the effects of a transversely moving boundary on plane Poiseuille flow. J. Biomech. Engng 104, 314–323.Google Scholar
Schumann, U. 1975 Linear stability of finite difference equations for three-dimensional flow problems. J. Comput. Phys. 18, 465–470.Google Scholar
Smith, F. T. 1986 Two-dimensional disturbance travel, growth and spreading in boundary layers. J. Fluid Mech. 169, 353–377.Google Scholar
Smith, F. T. & Bodonyi, R. J. 1985 On short-scale inviscid instabilities in flow past surfacemounted obstacles and other non-parallel motions. Aero. J. June/July, 205–212.Google Scholar
Smith, F. T. & Burggraf, O. R. 1985 On the development of large-sized, short-scaled disturbances in boundary layers. Proc. R. Soc. Lond. A 399, 25–55.Google Scholar
Sobey, I. J. 1980 On flow through furrowed channels. Part 1. Calculated flow patterns. J. Fluid Mech. 96, 1–26.Google Scholar
Sobey, I. J. 1985 Observation of waves during oscillatory channel flow. J. Fluid Mech. 151, 395–426.Google Scholar
Stephanoff, K. D., Pedley, T. J., Lawrence, C. J. & Secomb, T. W. 1983 Fluid flow along a channel with an asymmetric oscillating constriction. Nature 305, 692–695.Google Scholar
Stern, M. E. & Pratt, L. J. 1985 Dynamics of vorticity fronts. J. Fluid Mech. 161, 513–532.Google Scholar
Tutty, O. R. & Cowley, S. J. 1986 On the stability and the numerical solution of the unsteady interactive boundary-layer equation. J. Fluid Mech. 168, 431–456.Google Scholar
Viecelli, J. A. 1971 A computing method for incompressible flows bounded by moving walls. J. Comput. Phys. 8, 119–143.Google Scholar