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First- and higher-order effects of curvature and torsion on the flow in a helical rectangular duct

Published online by Cambridge University Press:  26 April 2006

C. Jonas Bolinder
C. Jonas Bolinder
Affiliation:
Division of Fluid Mechanics, Lund Institute of Technology, Box 118, 221 00 Lund, Sweden
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Abstract

A series expansion method is employed to determine the first-order terms in curvature ε and torsion η of fully developed laminar flow in helical square ducts and in helical rectangular ducts of aspect ratio two. The first-order solutions are compared to solutions of the full governing equations. For toroidal square ducts with zero pitch, the first-order solution is fairly accurate for Dean numbers, De = Re ε1/2, up to about 20, and for straight twisted square ducts the first-order solution is accurate for Germano numbers, Gn = η Re, up to at least 50 where Re is the Reynolds number. Important conclusions are that the flow in a helical duct with a finite pitch or torsion to the first order (i.e. with higher-order terms in ε and η neglected) is obtained as a superposition of the flow in a toroidal duct with zero pitch and a straight twisted duct; that the secondary flow in helical non-circular ducts for sufficiently small Re is dominated by torsion effects; and that for increasing Re, the secondary flow eventually is dominated by effects due to curvature. Torsion has a stronger impact on the flow for aspect ratios greater than one. A characteristic combined higher-order effect of curvature and torsion is an enlargement of the lower vortex of the secondary flow at the expense of the upper vortex, and also a shift of the maximum axial flow towards the upper wall. For higher Reynolds numbers, bifurcation phenomena appear. The extent of a few solution branches for helical ducts with finite pitch or torsion is determined. For ducts with small torsion it is found that the extent of the stable solution branches is affected little by torsion. Physical velocity components are employed to describe the flow. The contravariant components are found useful when describing the convective transport in the duct.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 314 , 10 May 1996 , pp. 113 - 138
Copyright
© 1996 Cambridge University Press

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References

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. 1978 Bifurcation phenomena in steady flows of a viscous fluid. Proc. R. Soc. Lond. A 359, 1–43.Google Scholar
Berger, S. A. 1991 Flow and heat transfer in curved pipes and tubes. AIAA Paper 91-0030.
Berger, S. A., Talbot, L. & Yao, L.-S. 1983 Flow in curved pipes. Ann. Rev. Fluid Mech. 15, 461–512.Google Scholar
Bounder, C. J. 1993 Numerical visualization of the flow in a helical duct of rectangular cross-section. Third Symp. on Experimental and Numerical Flow Visualization, New Orleans. ASME FED-Vol. 172, pp. 329–338.
Bolinder, C. J. 1995a Observations and predictions of laminar flow and heat transfer in helical rectangular ducts. Doctoral thesis, Lund Institute of Technology, Sweden.
Bolinder, C. J. 1995b The effect of torsion on the bifurcation structure of laminar flow in a helical square duct. Trans. ASME: J. Fluids Engng 117, 242–248.Google Scholar
Bolinder, C. J. 1996 Curvilinear coordinates and physical components – an application to the problem of viscous flow and heat transfer in smoothly curved ducts. Trans. ASME: J. Appl. Mech. (accepted).Google Scholar
Bolinder, C. J. & Sundén, B. 1995 Flow visualization and LDV measurements of laminar flow in a helical square duct with a finite pitch. Exp. Thermal Fluid Sci. 11, 348–363.Google Scholar
Bolinder, C. J. & Sundén, B. 1996 Numerical prediction of laminar flow and forced convective heat transfer in a helical square duct with a finite pitch. Intl J. Heat Mass Transfer (accepted).Google Scholar
Chen, W.-H. & Jan, R. 1992 The characteristics of laminar flow in a helical circular pipe. J. Fluid Mech. 244, 241–256.Google Scholar
Chen, W.-H. & Jan, R. 1993 The torsion effect on fully developed laminar flow in helical square ducts. Trans. ASME: J. Fluids Engng 115, 292–301.Google Scholar
Cheng, K. C., Lin, R.-C. & Ou, J.-W. 1976 Fully developed laminar flow in curved rectangular channels. Trans. ASME: J. Fluids Engng 98, 41–48.Google Scholar
Cuming, H. G. 1952 The secondary flow in curved pipes. Aeronaut. Res. Counc. RM 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 a fluid in a curved pipe. Phil. Mag. 4, 208–223.Google Scholar
Dean, W. R. 1928a The stream-line motion of fluid in a curved pipe. Phil. Mag. 5, 673–695.Google Scholar
Dean, W. R. 1928b Fluid motion in a curved channel. Proc. R. Soc. Lond. A 121, 402–420.Google Scholar
Duh, T.-Y. & Shih, Y.-D. 1989 Fully developed flow in curved channels of square cross sections inclined. Trans. ASME: J. Fluids Engng 111, 172–177.Google Scholar
Germano, M. 1982 On the effect of torsion on a helical pipe flow. J. Fluid Mech. 125, 1–8.Google Scholar
Germano, M. 1989 The Dean equations extended to a helical pipe flow. J. Fluid Mech. 203, 289–305.Google Scholar
Ghia, K. N., Ghia, U. & Shin, C. T. 1987 Study of fully developed incompressible flow in curved ducts, using a multi-grid technique. Trans. ASME: J. Fluids Engng 109, 226–236.Google Scholar
Ito, H. 1951 Theory on laminar flows through curved pipes of elliptic and rectangular cross-sections. Rep. Inst. High Speed Mech., vol. 1. Tohoku University, Sendai, Japan.
Ito, H. 1987 Flow in curved pipes. JSME 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
Kao, H. C. 1987 Torsion effect on fully developed flow in a helical pipe. J. Fluid Mech. 184, 335–356.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
Kheshgi, H. S. 1993 Laminar flow in twisted ducts. Phys. Fluids A 5, 2669–2681.Google Scholar
Liu, S. & Masliyah, J. H. 1993 Axially invariant laminar flow in helical pipes with a finite pitch. J. Fluid Mech. 251, 315–353.Google Scholar
Liu, S. & Masliyah, J. H. 1994 Developing convective heat transfer in helical pipes with finite pitch. Intl J. Heat Fluid Flow 15, 66–74.Google Scholar
Masliyah, J. H. & Nandakumar, K. 1981 Steady laminar flow through twisted pipes. Trans. ASME: J. Heat Transfer 103, 785–796.Google Scholar
Matsson, O. J. E. & Alfredsson, P. H. 1990 Curvature- and rotation-induced instabilities in channel flow. J. Fluid Mech. 210, 537–563.Google Scholar
Mees, P. A. J. 1994 Instability and transitions of flow in a curved duct of square cross section. Doctoral thesis, University of Alberta, Canada.
Mori, Y., Uchida, Y. & Ukon, T. 1971 Forced convective heat transfer in a curved channel with a square cross section. Intl J. Heat Mass Transfer 14, 1787–1805.Google Scholar
Murata, S., Miyake, Y., Inaba, T. & Ogawa, H. 1981 Laminar flow in a helically coiled pipe. Bull. JSME 24, 355–362.Google Scholar
Nandakumar, K. & Masliyah, J. H. 1983 Steady laminar flow through twisted pipes: fluid flow and heat transfer in rectangular tubes. Chem. Eng. Commun. 21, 151–173.Google Scholar
Nandakumar, K. & Masliyah, J. H. 1986 Swirling flow and heat transfer in coiled and twisted pipes. In Advances in Transport Processes, vol. 4 (ed. A. S. Mujumdar & R. A. Mashelkar), pp. 49–112. Wiley.
Nandakumar, K., Mees, P. A. J. & Masliyah, J. H. 1993 Multiple, two-dimensional solutions to the Dean problem in curved triangular ducts. Phys. Fluids A 5, 1182–1187.Google Scholar
Patankar, S. V. 1980 Numerical Heat Transfer and Fluid Flow. McGraw-Hill.
Sankar, S. R., Nandakumar, K. & Masliyah, J. H. 1988 Oscillatory flows in coiled square ducts. Phys. Fluids 31, 1348–1359.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
Sokolnikoff, I. S. 1964 Tensor Analysis. Wiley.
Thangam, S. & Hur, N. 1990 Laminar secondary flows in curved rectangular ducts. J. Fluid Mech. 217, 421–440.Google Scholar
Timoshenko, S. P. & Woinowsky-Krieger, S. 1959 Theory of Plates and Shells. McGraw-Hill.
Todd, L. 1986 Steady laminar flow through thin curved pipes. Fluid Dyn. Res. 1, 237–255.Google Scholar
Tuttle, E. R. 1990 Laminar flow in twisted pipes. J. Fluid Mech. 219, 545–570.Google Scholar
Van Doormaal, J. P. & Raithby, G. D. 1984 Enhancements of the SIMPLE method for predicting incompressible fluid flows. Numer. Heat Transfer 7, 147–163.Google Scholar
Van Dyke, M. 1978 Extended Stokes series: laminar flow through a loosely coiled pipe. J. Fluid Mech. 86, 129–145.Google Scholar
Wang, C. Y. 1981 On the low-Reynolds-number flow in a helical pipe. J. Fluid Mech. 108, 185–194.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
Xie, D. G. 1990 Torsion effect on secondary flow in a helical pipe. Intl J. Heat Fluid Flow 11, 114–119.Google Scholar
Yanase, S., Goto, N. & Yamamoto, K. 1989 Dual solutions of the flow through a curved tube. Fluid Dyn. Res. 5, 191–201.Google Scholar
Yanase, S. & Nishiyama, K. 1988 On the bifurcation of laminar flows through a curved rectangular tube, J. Phys. Soc. Japan 57, 3790–3795.Google Scholar
Yang, Z.-H. & Keller, H. B. 1986 Multiple laminar flows through curved pipes. Appl. Num. Math. 2, 257–271.Google Scholar