Hostname: page-component-77c89778f8-vsgnj Total loading time: 0 Render date: 2024-07-23T21:17:33.293Z Has data issue: false hasContentIssue false
  • English
  • Français

A bifurcation study of viscous flow through a rotating curved duct

Published online by Cambridge University Press:  26 April 2006

M. Selmi ,
K. Nandakumar  and
W. H. Finlay
M. Selmi
Affiliation:
Mechanical Engineering Department, Qatar University, PO Box 2713, Doha - Qatar
K. Nandakumar
Affiliation:
Department of Chemical Engineering, University of Alberta, Edmonton, Alberta, Canada, T6G 2G6
W. H. Finlay
Affiliation:
Department of Mechanical Engineering, University of Alberta, Edmonton, Alberta, Canada, T6G 2G8
Get access Rights & Permissions [Opens in a new window]

Abstract

The combined effects of system rotation (Coriolis force) and curvature (centrifugal force) on the bifurcation structure of two-dimensional flows in a toroidal geometry of rectangular cross-section are examined. The problem depends on the Rossby number, Ro = U/bΩ, the Ekman number, Ek = ν/b2Ω, the aspect ratio, γ = b/h and the radius ratio, η = ri/ro; here U is the velocity scale, b is the channel width in the spanwise direction, Ω is the rotational speed, (ri, ro) are the inner and outer radii of the duct, h = ro – ri is the channel gap in the radial direction and v is the kinematic viscosity of the fluid. A pseudospectral method is devised to discretize the two-dimensional Navier–Stokes equation in stream-function form. Continuation schemes are used to track the solution paths with Rossby number as the control parameter. Extended systems are used to determine the precise location of the singular points of the discretized system. The loci of such singular points are tracked with respect to curvature of the duct. Unlike the findings of Miyazaki (1973) on the same problem, curvature is found to have profound effects on the solution structure; flow mutations take place through a tilted cusp at (Ro = 7.122, η = 0.678) and a transcritical bifurcation point at (Ro = 1.357, η = 0.349).

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 262 , 3 October 1994 , pp. 353 - 375
Copyright
© 1994 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

Alfredsson, P. A. & Persson, H. 1989 Instabilities in channel flow with system rotation. J. Fluid Mech. 202, 543–557.Google Scholar
Bara, B., Nandakumar, K. & Masliyah, J. H. 1992 An experimental and numerical study of the Dean problem: flow development towards two-dimensional multiple solutions. J. Fluid Mech. 244, 339–376.Google Scholar
Barua, S. N. 1954 On secondary flow in stationary curved pipes. Proc. R. Soc. Lond. A 227, 133–139.Google Scholar
Berger, S. A., Talbot, L. & Yao, L-S. 1983 Flow in curved pipes. Ann. Rev. Fluid Mech. 15, 461–512.Google Scholar
Cheng, K. C. & Akiyama, M. 1970 Laminar forced convection heat transfer in curved rectangular channels. Intl J. Heat Mass Transfer 13, 471–490.Google Scholar
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 1977. Hemisphere.
Daskopoulos, P. & Lenhoff, A. M. 1989 Flow in curved ducts: bifurcation structure for stationary ducts. J. Fluid Mech. 203, 125–148.Google Scholar
Daskopoulos, P. & Lenhoff, A. M. 1990 Flow in curved ducts. Part 2. Rotating ducts. J. Fluid Mech. 217, 575–593.Google Scholar
Dean, W. R. 1927 Note on the motion of fluid in a curved pipe. Phil. Mag. 4, 208–223.Google Scholar
Dennis, S. R. C. 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. R. C. & Riley, N. 1982 On the fully developed flow in a curved pipe at large Dean number. Proc. R. Soc. Lond. A 434, 473–478.Google Scholar
Duck, P. 1983 Flow through rotating straight pipes of a circular cross section. Phys. Fluids 26, 614–618.Google Scholar
Durst, F. & Raszillier, H. 1990 Flow in a rotating straight pipe with a view on Coriolis mass flow meters. Trans. ASME I J. Fluids Engng 112, 149–154.Google Scholar
Eustice, J. 1910 Flow of water in curved pipes. Proc. R. Soc. Lond. A 84, 107–118.Google Scholar
Eustice, J. 1911 Experiments on stream-line motion in curved pipes. Proc. R. Soc. Lond. A 85, 119–131.Google Scholar
Finlay, W. H. 1990 Transition to oscillatory motion in rotating channel flow. J. Fluid Mech. 215, 209–227.Google Scholar
Finlay, W. H. 1992 Transition to turbulence in a rotating channel. J. Fluid Mech. 237, 73–99.Google Scholar
Ghia, K. N. & Sokhey, J. S. 1977 Laminar incompressible viscous flow in curved ducts of regular cross-section. Trans. ASME I J. Fluids Engng 99, 640–648.Google Scholar
Hart, J. E. 1971 Instability and secondary motion in a rotating channel flow. J. Fluid Mech. 45, 341–351.Google Scholar
Hille, P., Vehrenkamp, R. & Schulz-Dubois, E. O. 1985 The development and structure of primary and secondary flow in a curved square duct. J. Fluid Mech. 151, 219–241.Google Scholar
Hocking, L. M. 1967 Boundary and shear layers in a curved rotating pipe. J. Math. Phys. Sci. 1, 123–136.Google Scholar
Humphrey, J. A. C., Taylor, A. M. K. & Whitelaw, J. H. 1977 Laminar flow in a square duct of strong curvature. J. Fluid Mech. 83, 509–527.Google Scholar
Itō, H. & Motai, T. 1974 Secondary flow in a rotating curved pipe. Rep. Inst. High Speed Mech. 29, 33–57.Google Scholar
Kheshgi, H. S. & Scriven, L. E. 1985 Viscous flow through a rotating square channel. Phys. Fluids 28, 2868–2979.Google Scholar
Lezius, D. K. & Johnston, J. P. 1976 Roll-cell instabilities in rotating laminar and turbulent channel flows. J. Fluid Mech. 77, 153–175.Google Scholar
Ludwieg, H. 1951 Die ausgebildete Kanalströmung in einem rotierenden System. Ingenieur-Archiv 19, 296–308.Google Scholar
Mansour, K. 1985 Laminar flow through a slowly rotating straight pipe. J. Fluid Mech. 150, 1–21.Google Scholar
Miyazaki, H. 1971 Combined free and forced convective heat transfer and fluid flow in a rotating curved circular tube. Intl J. Heat Mass Transfer 14, 1295–1309.Google Scholar
Miyazaki, H. 1973 Combined free and forced convective heat transfer and fluid flow in a rotating curved rectangular tube. Trans. ASME C: J. Heat Transfer 95, 64–71.Google Scholar
Moore, G. & Spence, A. 1980 The calculation of turning points of non-linear equations. SIAM J. Numer. Anal. 17, 567–576.Google Scholar
Nandakumar, K. & Masliyah, J. H. 1986 Swirling flow and heat transfer in coiled and twisted pipes. Adv. Transport Processes 4, 49–112.Google Scholar
Nandakumar, K., Raszillier, H. & Durst, F. 1991 Flow through rotating rectangular ducts. Phys. Fluids A 3, 770–781.Google Scholar
Nandakumar, K. & Weinitschke, H. J. 1991 A bifurcation study of mixed convection heat transfer in horizontal ducts. J. Fluid Mech. 231, 157–187.Google Scholar
Piesche, M. & Felsh, K.-O. 1980 Experimental investigation of pressure loss in rotating curved rectangular channels. Arch. Mech. 32, 747–756.Google Scholar
Ramshankar, R. & Sreenivasan, K. R. 1988 A paradox concerning the extended Stokes series solution for the pressure drop in coiled pipes. Phys. Fluids 31, 1339–1347.Google Scholar
Raszillier, H. & Durst, F. 1991 Coriolis-effect in mass flow metering. Arch. Appl. Mech. 61, 192–214.Google Scholar
Sankar, S. R., Nandakumar, K. & Masliyah, J. H. 1988 Oscillatory flows in coiled square ducts. Phys. Fluids 31, 1348–1358.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
Spence, A. & Werner, B. 1982 Non-simple turning points and cusps. IMA J. Numer. Anal. 2, 413–427.Google Scholar
Speziale, C. G. 1982 Numerical study of viscous flow in rotating rectangular ducts. J. Fluid Mech. 122, 251–271.Google Scholar
Speziale, C. G. & Thangam, S. 1983 Numerical study of secondary flows and roll-cell instabilities in rotating channel flow. J. Fluid Mech. 130, 377–395.Google Scholar
Sugiyama, S., Hayashi, T. & Yamazaki, K. 1983 Flow characteristics in the curved rectangular channels (visualization of secondary flow). Bull. JSME 26, 964–969.Google Scholar
Taylor, A. M. K. P., Whitelaw, J. H. & Yianneskis, M. 1982 Curved ducts with strong secondary motion: velocity measurements of developing laminar and turbulent flow. Trans. ASME I: J. Fluids Engng 104, 350–359.Google Scholar
Van Dyke, M. 1978 Extended stokes series: laminar flow through a loosely coiled pipe. J. Fluid Mech. 86, 129–145.Google Scholar
Walker, J. S. 1975 Steady flow in a rapidly rotating variable area rectangular ducts J. Fluid Mech. 69, 209–227.Google Scholar
Werner, B. & Spence, A. 1984 The computation of symmetry breaking bifurcation points. SIAM J. Numer. Anal. 21, 388–399.Google Scholar
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.
Yee, G., Chilukuri, R. & Humphery, J. A. C. 1980 Developing flow and heat transfer in strongly curved ducts of rectangular cross section. Trans. ASME C: J. Heat Transfer 102, 285–291.Google Scholar