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Bifurcation in steady laminar flow through curved tubes

Published online by Cambridge University Press:  20 April 2006

K. Nandakumar  and
Jacob H. Masliyah
K. Nandakumar
Affiliation:
Department of Chemical Engineering, University of Alberta, Edmonton, Alberta, Canada
Jacob H. Masliyah
Affiliation:
Department of Chemical Engineering, University of Alberta, Edmonton, Alberta, Canada
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Abstract

The occurrence of dual solutions in curved ducts is investigated through a numerical solution of the Navier-Stokes equations in a bipolar-toroidal co-ordinate system. With the shape of duct being the region formed by the natural co-ordinate surfaces, it was possible to alter the duct geometry gradually and preserve the prevailing form of the velocity field, in a manner suggested by Benjamin (1978).

In addition to the Dean number Dn = Re/Rc½, a geometrical parameter that defines the shape of the duct was also varied systematically to study the bifurcation of a two-vortex solution into a two- and four-vortex solution. Dual solutions have been found for all geometrical shapes investigated here. Of particular interest are the shapes of a full circle and a semicircle with a curved outer wall.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 119 , June 1982 , pp. 475 - 490
Copyright
© 1982 Cambridge University Press

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References

Austin, L. R. & Seader, J. D. 1973 Fully developed viscous flow in coiled circular pipes. A.I.Ch.E. J. 19, 85–94.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; II. Experiments. Proc. R. Soc. Lond. A 359, 27–43.Google Scholar
Brandt, A. 1980 Multilevel adaptive computations in fluid dynamics. A.I.A.A. J. 18, 1165–1172.Google Scholar
Cheng, K. C. & Akiyama, M. 1970 Laminar forced convection heat transfer in curved rectangular channels. Int. J. Heat Mass Transfer 13, 471–490.Google Scholar
Cheng, K. C., Lin, R.-C. & Ou, J.-W. 1976 Fully developed laminar flow in curved rectangular channels. Trans. A.S.M.E.I., J. Fluids Engng 98, 41–48.Google Scholar
Collins, W. M. & Dennis, S. C. R. 1975 The steady motion of a viscous fluid in a curved tube. Quart. J. Mech. Appl. Math. 28, 133–156.Google Scholar
Collins, W. M. & Dennis, S. C. R. 1976a Viscous eddies near a 90 and a 45 corner in flow through a curved tube of triangular cross-section. J. Fluid Mech. 76, 417–432.Google Scholar
Collins, W. M. & Dennis, S. C. R. 1976b Steady flow in a curved tube of triangular cross section. Proc. R. Soc. Lond. A 352, 189–211.Google Scholar
Cuming, H. G. 1952 The secondary flow in curved pipes. Aero. Res. Counc. R. & M. no. 2880.Google Scholar
Dean, W. R. 1928 The stream-line motion of fluid in a curved pipe. Phil. Mag. 5(7), 673–695.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. C. 1982 Dual solution for steady laminar flow through a curved tube. Q. J. Mech. Appl. Math. (in press).
De Vriend, H. J. 1981 Velocity redistribution in curved rectangular channels. J. Fluid Mech. 107, 423–439.Google Scholar
Joseph, B., Smith, E. P. & Adler, R. J. 1975 Numerical treatment of laminar flow in helically coiled tubes of square cross section. A.I.Ch.E. J. 21, 965–974.Google Scholar
Lin, J. Y. & Tarbell, J. M. 1980 An experimental and numerical study of periodic flow in a curved tube. J. Fluid Mech. 100, 623–638.Google Scholar
Manlapaz, R. L. & Churchill, S. W. 1980 Fully developed laminar flow in a helically coiled tube of finite pitch. Chem. Engng Commun. 7, 57–78.Google Scholar
Masliyah, J. H. 1980 On laminar flow in curved semicircular ducts. J. Fluid Mech. 99, 469–479.Google Scholar
Masliyah, J. H. & Nandakumar, K. 1979 Fully developed viscous flow and heat transfer in curved semi-circular sectors. A.I.Ch.E. J. 25, 478–487.Google Scholar
Nandakumar, K. & Masliyah, J. H. 1981 Laminar flow in curved moon-shaped ducts. In Proc. 2nd Int. Conf. on Numerical Methods in Laminar and Turbulent Flow, Venice (Ed. C. Taylor & B. A. Schrefler). Pineridge.
Ng, M. C. 1980 Steady fully developed flow through a curved tube of circular cross-section. 4th Canadian Symp. on Fluid Dynamics, Abstract, p. 48.
Tarbell, J. M. & Samuels, M. R. 1973 Momentum and heat transfer in helical coils. Chem. Engng J. 5, 117–127.Google Scholar
Truesdell, L. C. & Adler, R. J. 1970 Numerical treatment of fully developed laminar flow in helically coiled tubes. A.I.Ch.E. J. 16, 1010–1015.Google Scholar
Van Dyke, M. 1978 Extended Stokes series: laminar flow through a loosely coiled pipe. J. Fluid Mech. 86, 129–145.Google Scholar