Hostname: page-component-77c89778f8-cnmwb Total loading time: 0 Render date: 2024-07-17T10:34:49.158Z Has data issue: false hasContentIssue false
  • English
  • Français

Instability and transitions of flow in a curved square duct: the development of two pairs of Dean vortices

Published online by Cambridge University Press:  26 April 2006

Philip A. J. Mees ,
K. Nandakumar  and
J. H. Masliyah
Philip A. J. Mees
Affiliation:
Department of Chemical Engineering, University of Alberta, Edmonton, Alberta, Canada T6G 2G6
K. Nandakumar
Affiliation:
Department of Chemical Engineering, University of Alberta, Edmonton, Alberta, Canada T6G 2G6
J. H. Masliyah
Affiliation:
Department of Chemical Engineering, University of Alberta, Edmonton, Alberta, Canada T6G 2G6
Get access Rights & Permissions [Opens in a new window]

Abstract

Steady developing flow of an incompressible Newtonian fluid in a curved duct of square cross-section (the Dean problem) is investigated both experimentally and numerically. This study is a continuation of the work by Bara, Nandakumar & Masliyah (1992) and is focused on flow rates between Dn = 200 and Dn = 600 (Dn = Re/(R/a)1/2, where Re is the Reynolds number, R is the radius of curvature of the duct and a is the duct dimension; the curvature ratio, R/a, is 15.1).

Numerical simulations based on the steady three-dimensional Navier – Stokes equations predict the development of a 6-cell secondary flow pattern above a Dean number of 350. The 6-cell state consists of two large Ekman vortices and two pairs of small Dean vortices near the outer wall that result from the primary instability that is of centrifugal nature. The 6-cell flow state develops near θ = 80° and breaks down symmetrically into a 2-cell flow pattern.

The apparatus used to verify the simulations had a duct dimension of 1.27 cm and a streamwise length of 270°. At a Dean number of 453, different velocity profiles of the 6-cell flow state at θ = 90° and spanwise profiles of the streamwise velocity at every 20° were measured using a laser-Doppler anemometer. All measured velocity profiles, as well as flow visualization of secondary flow patterns, are in very good agreement with the simulations, indicating that the parabolized Navier – Stokes equations give an accurate description of the flow.

Based on the similarity with boundary layer flow over a concave wall (the Görtler problem), it is suggested that the transition to the 6-cell flow state is the result of a decreasing spanwise wavelength of the Dean vortices with increasing flow rate. A numerical stability analysis shows that the 6-cell flow state is unconditionally unstable. This is the first time that detailed experiments and simulations of the development of a 6-cell flow state are reported.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 314 , 10 May 1996 , pp. 227 - 246
Copyright
© 1996 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

Adler, M. 1934 Strömung in gekrümmten Rohren. Z. Angew. Math. Mech. 14, 257–275.Google Scholar
Arnal, M. P., Goering, D. J. & Humphrey, J. A. C. 1992 Unsteady laminar flow developing in a curved duct. Intl J. Heat Fluid Flow 13, 347–357.Google Scholar
Bara, B. 1991 Experimental investigation of developing and fully developed flow in a curved duct of square cross section. PhD thesis, Department of Mechanical Engineering, University of Alberta.
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
Benjamin, T. B. 1978a Bifurcation phenomena in steady flows of a viscous fluid, I. Theory. Proc. R. Soc. Lond. A 359, 1–26.Google Scholar
Benjamin, T. B. 1978b Bifurcation phenomena in steady flows of a viscous fluid, II. Eperiments. Proc. R. Soc. Lond. A 359, 27–43.Google Scholar
Berger, S. A., Talbot, L. & Yao, L.-S. 1983 Flow in curved pipes. Ann. Rev. Fluid Mech. 15, 461–512.Google Scholar
Bippes, H. 1972 Experimentelle Untersuchung des laminar-turbulenten Umschlags an einer parallel angeströmten konkaven Wand. Sitzungsber. Heidel. Akad. Wiss., Math.-naturwiss. Kl. 3, 103–180. Also NASA Tech. Mem. 75243 (1978).Google Scholar
Bippes, H. & Görtler, H. 1972 Dreidimensionale Störungen in der Grenzschicht an einer konkaven Wand. Acta Mech. 14, 251–267.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., Lin, R.-C. & Ou, J.-W. 1975 Graetz problem in curved square channels. Trans. ASME C: J. Heat Transfer 97, 244–248.Google Scholar
Cheng, K. C., Lin, R.-C. & Ou, J.-W. 1976 Fully developed flow in curved rectangular channels. Trans. ASME I: J. Fluids Engng. 98, 41–48.Google Scholar
Cuming, H. G. 1952 The secondary flow in curved pipes. Aeronaut. Res. Counc. Rep. Mem. 2880.
Daskopoulos, P. & Lenhoff, A. M. 1989 Flow in curved ducts: bifurcation structure for stationary ducts. J. Fluid Mech. 203, 125–148.Google Scholar
Dean, W. R. 1927 Note on the motion of fluid in a curved pipe. Phil. Mag. (7) 4, 208–223.Google Scholar
Dean, W. R. 1928 The stream-line motion of fluid in a curved pipe. Phil. Mag. (7) 5, 673–695.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
Eustice, J. 1925 Flow of fluids in curved passages. Engineering, November 13, 604–605.
Finlay, W. H., Guo, Y. & Olsen, D. 1993 Inferring secondary flows from smoke or dye flow visualization: Two case studies. Phys. Fluids A 5, 2689–2701.Google Scholar
Finlay, W. H. & Nandakumar, K. 1990 Onset of two-dimensional cellular flow in finite curved channels of large aspect ratio. Phys. Fluids A 2, 1163–1174.Google Scholar
Floryan, J. M. & Saric, W. S. 1984 Wavelength selection and growth of Görtler vortices. AIAA J. 22, 1529–1538.Google Scholar
Ghia, K. N. & Sokhey, J. S. 1977 Laminar incompressible viscous flow in curved ducts of regular cross-sections. Trans. ASME I: J. Fluids Engng 99, 640–648.Google Scholar
Goldstein, R. J. & Kreid, D. K. 1967 Measurement of laminar flow development in a square duct using a laser-Doppler flowmeter. Trans. ASME E: J. Appl. Mech. 34, 813–818.Google Scholar
Guo, Y. & Finlay, W. H. 1994 Wavenumber selection and irregularity of spatially developing nonlinear Dean and Görtler vortices. J. Fluid Mech. 264, 1–40.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
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. 1951 Theory on laminar flows through curved pipes of elliptic and rectangular cross-sections. Rep. Inst. High Speed Mech., Tōhoku Univ., Sendai, Japan 1, 1–16.
Itō, H. 1959 Friction factors for turbulent flow in curved pipes. Trans. ASME D: J. Basic Engng 81, 123–134.Google Scholar
Itō, H. 1987 Flow in curved pipes. JSME Intl J. II 30, 543–552.Google Scholar
Joseph, B., Smith, E. P. & Alder, R. J. 1975 Numerical treatment of laminar flow in helically coiled tubes of square cross section. Part 1. Stationary helically coiled tubes. AIChE J. 21, 965–974.Google Scholar
Kajishima, T., Miyake, Y. & Inaba, T. 1989 Numerical simulation of laminar flow in curved ducts of rectangular cross-section. JSME Intl J. II 32, 516–522.Google Scholar
Kao, H. C. 1992 Some aspects of bifurcation structure of laminar flow in curved ducts. J. Fluid Mech. 243, 519–539.Google Scholar
Liu, S., Afacan, A., Nasr-el-Din, H. A. & Masliyah, J. H. 1994 An experimental study of pressure drop in helical pipes. Proc. R. Soc. Lond. A 444, 307–316.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
Manlapaz, R. L. & Churchill, S. W. 1981 Fully developed laminar convection from a helical coil. Chem. Engng Commun. 9, 185–200.Google Scholar
Mees, P. A. J. 1994 Instability and transitions of flow in a curved duct of square cross section. PhD thesis, Department of Chemical Engineering, University of Alberta.
Mees, P. A. J., Nandakumar, K. & Masliyah, J. H. 1996 Steady spatial oscillations in a curved duct of square cross section (submitted for publication).
Miyake, Y., Kajishima, T. & Inaba, T. 1988 Numerical experiment of laminar flow in curved ducts of rectangular cross-section. In Experimental Heat Transfer, Fluid Mechanics, and Thermodynamics: Proc. First World Conf. on Experimental Heat Transfer, Fluid Mechanics, and Thermodynamics, Sept. 4–9, Dubrovnik, Yugoslavia (ed. R. K. Shah, E. N. Ganić & K. T. Yand), pp. 1192–1199.
Mori, Y. & Nakayama, W. 1965 Study on forced convective heat transfer in curved pipes. Part 1. Intl J. Heat Mass Transfer 8, 67–82.Google Scholar
Nandakumar, K. & Masliyah, J. H. 1986 Swirling flow and heat transfer in coiled and twisted pipes. In Advances in Transport Processe (ed. A. S. Mujumdar & R. A. Mashelkar), pp. 49–112. New Delhi: Wiley Eastern.
Patankar, S. V. 1980 Numerical Heat Transfer and Fluid Flow. Washington, DC: Hemisphere.
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
Sankar, S. R., Nandakumar, K. & Masliyah, J. H. 1988 Oscillatory flows in coiled square ducts. Phys. Fluids 31, 1348–1359.Google Scholar
Shah, R. K. & London, A. L. 1978 Laminar Flow Forced Convection in Ducts. Academic.
Shantini, 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
Sugiyama, S., Aoi, T., Yamamoto, M., Narisawa, N. & Miyake, Y. 1988 Measurements on developing laminar flow in a curved rectangular duct by means of LDV. In Experimental Heat Transfer, Fluid Mechanics, and Thermodynamics: Proc. First World Conf. on Experimental Heat Transfer, Fluid Mechanics, and Thermodynamics, Sept. 4–9, Dubrovnik, Yugoslavia (ed. R. K. Shah, E. N. Ganić, & K. T. Yang), pp. 1185–1191.
Sugiyama, S., Hayashi, T. & Yamazaki, K. 1983 Flow characteristics in the curved rectangular channels. 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
Thangam, S. & Hur, N. 1990 Laminar secondary flows in curved rectangular ducts. J. Fluid Mech. 217, 421–440.Google Scholar
Van Dyke, M. 1978 Extended Stokes series: laminar flow through a loosely coiled pipe. J. Fluid Mech. 86, 129–145.Google Scholar
Williams, G. S., Hubbell, C. W. & Finkell, G. H. 1902 Experiments at Detroit, Michigan on the effect of curvature on the flow of water in pipes. Trans. ASCE 47, 1–196.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
Yao, L.-S. & Berger, S. A. 1988 The three-dimensional boundary layer in the entry region of curved pipes with finite curvature ratio. Phys. Fluids 31, 486–494.Google Scholar
Yee, G., Chilukuri, R. & Humphrey, 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