Hostname: page-component-745bb68f8f-b95js Total loading time: 0 Render date: 2025-01-09T19:19:09.178Z Has data issue: false hasContentIssue false
  • English
  • Français

Flow in curved ducts: bifurcation structure for stationary ducts

Published online by Cambridge University Press:  26 April 2006

P. Daskopoulos  and
A. M. Lenhoff
P. Daskopoulos
Affiliation:
Department of Chemical Engineering, University of Delaware, Newark, DE 19716, USA
A. M. Lenhoff
Affiliation:
Department of Chemical Engineering, University of Delaware, Newark, DE 19716, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

In developed laminar flow in stationary curved ducts there is, in addition to the two-vortex secondary flow structure first analysed by Dean (1927), another solution branch with a four-vortex secondary flow. These two branches have recently been shown to be joined by a third branch, but stability characteristics and the possible presence of additional branches have yet to be described. In this paper orthogonal collocation is used in conjunction with continuation techniques to characterize the bifurcation structure for symmetric flows. The two- and four-vortex solutions are stable to symmetric disturbances, while the recently reported branch joining them is unstable. A more systematic exploration of the parameter space than has hitherto been reported is performed by examining the morphogenesis of the bifurcation structure within the general framework of properties described by Benjamin (1978). The starting point is the ‘perfect’ problem of flow in an infinite curved slit, which bifurcates to give rise to a cellular structure. Addition of ‘stickiness’ at the cell boundaries turns each pair of cells into a curved duct of rectangular cross-section which, by a geometry change, leads to the curved circular tube. For the perfect problem a large number of solution branches are present, but the addition of stickiness turns most of them into isolae which vanish before the no-slip limit is reached. The solution branches that remain include, in addition to the three described previously, another solution family not connected to the other one. This family comprises two branches, both four-vortex in character and unstable to symmetric disturbances.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 203 , June 1989 , pp. 125 - 148
Copyright
© 1989 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

Barua, S. N.: 1963 On secondary flow in stationary curved pipes. Q. J. Mech. Appl. Maths 16, 61–77.Google Scholar
Benjamin, T. B.: 1978 Bifurcation phenomena in steady flows of a viscous fluid. I. Theory. Proc. R. Soc. Lond. A 359, 1–26.Google Scholar
Berger, S. A., Talbot, L. & Yao, L.-S. 1983 Flow in curved pipes. Ann. Rev. Fluid Mech. 15, 461–512.Google Scholar
Chandrasekhar, S.: 1961 Hydrodynamic and Hydromagnetic Stability, Chapter 8. Oxford University Press.
Cheng, K. C. & Akiyama, M., 1970 Laminar forced convection heat transfer in curved rectangular channels. Intl J. Heat Mass Trans. 13, 471–490.Google Scholar
Cheng, K. C., Inaba, T. & Akiyama, M., 1985 Flow visualization studies of secondary flow patterns and centrifugal instability in curved circular and semicircular pipes. In Flow Visualization III (ed. W. J. Yang), Third Intl Symp. on Flow Visualization, 1983, Ann Arbor, Michigan, pp. 531–536. Springer.
Cheng, K. C., Nakayama, J. & Akiyama, M., 1979 Effect of finite and infinite aspect ratios on flow patterns in curved rectangular channels. In Flow Visualization (ed. T. Asanuma), Intl Symp. on Flow Visualization, 1977, Tokyo, Japan, pp. 181–186. Hemisphere.
Collins, W. M. & Dennis, S. C. R. 1975 The steady motion of a viscous fluid in a curved tube. Q. J. Mech. Appl. Maths 28, 133–156.Google Scholar
Dean, W. R.: 1927 Note on the motion of fluid in a curved pipe. Phil. Mag. 4, (7) 208–223.Google Scholar
Dean, W. R.: 1928a The streamline motion of fluid in a curved pipe (second paper). Phil. Mag. 5, (7) 673–695.Google Scholar
Dean, W. R.: 1928b Fluid motion in a curved channel. Proc. R. Soc. Lond. A 121, 402–420.Google Scholar
Dennis, S. C. R.: 1980 Calculation of the steady flow through a curved tube using a new finite-difference method. J. Fluid Mech. 99, 449–467.Google Scholar
Dennis, S. C. R. & Ng, M. 1982 Dual solutions for steady laminar flow through a curved tube. Q. J. Mech. Appl. Maths 35, 305–324.Google Scholar
De Vriend, H. J. 1981 Velocity redistribution in curved rectangular channels. J. Fluid Mech. 107, 423–439.Google Scholar
Drazin, P. G. & Reid, W. H., 1981 Hydrodynamic Stability, Chapter 3. Cambridge University Press.
Gottlieb, D. & Orszag, S. A., 1977 Numerical Analysis of Spectral Methods: Theory and Applications. SIAM.
Itō, H.: 1987 Flow in curved pipes. Japan. Soc. Mech. Engrs Intl J. 30, 543–552.Google Scholar
Joseph, B., Smith, E. P. & Adler, R. J., 1975 Numerical treatment of laminar flow in helically coiled tubes of square cross section. AIChE J. 21, 965–974.Google Scholar
Keller, H. B.: 1977 Numerical solution of bifurcation and nonlinear eigenvalue problems. In Applications of Bifurcation Theory (ed. P. H. Rabinowitz), pp. 359–384. Academic.
Keller, H. B.: 1982 Continuation methods in computational fluid dynamics. In Numerical Methods and Physical Aspects of Aerodynamic Flows (ed. T. Cebeci), pp. 3–13. Springer.
Kubíĉek, M. & Marek, M. 1983 Computational Methods in Bifurcation Theory and Dissipative Structures. Springer.
Lanczos, C.: 1956 Applied Analysis, Chapter 7. Prentice-Hall.
Masliyah, J. H.: 1980 On laminar flow in curved semicircular ducts. J. Fluid Mech. 99, 469–479.Google Scholar
McConalogue, D. J. & Srivastava, R. S., 1968 Motion of a fluid in a curved tube. Proc. R. Soc. Lond. A 307, 37–53.Google Scholar
Nandakumar, K. & Masliyah, J. H., 1982 Bifurcation in steady laminar flow through curved tubes. J. Fluid Mech. 119, 475–490.Google Scholar
Nandakumar, K., Masliyah, J. H. & Law, H.-S. 1985 Bifurcation in steady laminar mixed convection flow in horizontal ducts. J. Fluid Mech. 152, 145–161.Google Scholar
Reid, W. H.: 1958 On the stability of viscous flow in a curved channel. Proc. R. Soc. Lond. A 244, 186–198.Google Scholar
Shanthini, W. & Nandakumar, K., 1986 Bifurcation phenomena of generalized newtonian fluids in curved rectangular ducts. J. Non-Newtonian Fluid Mech. 22, 35–60.Google Scholar
Soh, W. Y.: 1988 Developing fluid flow in a curved duct of square cross-section and its fully developed dual solutions. J. Fluid Mech. 188, 337–361.Google Scholar
Soh, W. Y. & Berger, S. A., 1987 Fully developed flow in a curved pipe of arbitrary curvature ratio. Intl J. Numer. Meth. Fluids 7, 733–755.Google Scholar
Taylor, G. I.: 1923 Stability of a viscous liquid contained between two rotating cylinders. Phil. Trans. R. Soc. Lond. A 223, 289–343.Google Scholar
Van Dyke, M. 1978 Extended Stokes series: laminar flow through a loosely coiled pipe. J. Fluid Mech. 86, 129–145.Google Scholar
Ward-Smith, A. J.: 1980 Internal Fluid Flow, Chapter E. Clarendon.
Winters, K. H.: 1987 A bifurcation study of laminar flow in a curved tube of rectangular cross-section. J. Fluid Mech. 180, 343–369.Google Scholar
Winters, K. H. & Brindley, R. C. G. 1984 Multiple solutions for laminar flow in helically-coiled tubes. AERE Rep. 11373, AERE Harwell, UK.
Yang, Z.-H. & Keller, H. B. 1986a Multiple laminar flows through curved pipes. Appl. Num. Maths 2, 257–271.Google Scholar
Yang, Z.-H., Keller, H. B.: 1986b Multiple laminar flows through curved pipes. In Proc. Tenth Intl Conf. on Numerical Methods in Fluid Dynamics, Beijing 1986 (ed. F. G. Zhuang, Y. L. Zhu), pp. 672–676. Springer.