Abstract
Laminar flows through 180° curved bends of circular cross section are investigated numerically. For small curvature ratio, α, defined as pipe radius over mean bend radius, the governing equations could be parabolized. The equations are solved for an α range of from 0.04 to 0.143, a Dean number (De) range of from 277.5 to 1360, and for a uniform flow, a potential vortex, and a parabolic flow inlet condition. In all these studies a zero cross-stream flow at the inlet is assumed. A detailed study of the effects of α, De, and inlet condition on the secondary flow pattern is carried out. Within the range of parameters investigated, up to three secondary cells are found in the cross-stream half-plane of a curved pipe. They are the Dean-type secondary cell, a secondary separation cell near the inner bend (closest to the center of curvature of the bend), and a third cell near the pipe center. The number of secondary cells in the cross-stream half-plane is greatly influenced by the inlet flow, and to a much lesser extent by α and De. For example, only the Dean cell is found in a curved-pipe flow where α and De are small and the inlet flow is either uniform or a potential vortex. When the inlet condition of the same case is changed to a parabolic flow, a three-cell structure results. Furthermore, as De increases to 1180, incipient axial flow separation begins at around 23° downstream of the curved-pipe entrance. The formation and extent of the separation and third cells are investigated together with their dependence on the parameters studied. This investigation further shows that, within the range of parameters examined, there is no secondary cell occurring near the outer bend, contrary to some earlier findings on fully developed curved-pipe flows.
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Agrawal, Y., Talbot, L., and Gong, K. (1978) Laser anemometer study of flow development in curved circular pipes. J. Fluid Mech. 85, 497–578.
Akiyama, H., Hanaoka, Y., Cheng, K.C., Uroi, I., and Suzuki, M. (1983) Visual measurements of laminar flow in the entry region of a curved pipe. Proc. 3rd Internat. Symp. on Flow Visualization, Ann Arbor (ed. W.J. Yang), pp. 526–530.
Anwer, M., So, R.M.C., and Lai, Y.G. (1989) Perturbation by and recovery from bend curvature of a fully developed turbulent pipe flow. Phys. Fluids A 1, 1387–1397.
Azzola, J., Humphrey, J.A.C., Iacovides, H., and Launder, B.E. (1986) Developing turbulent flow in a U-bend of circular cross-section: measurement and computation. J. Fluids Engrg. 43, 711–783.
Berger, S.A., Talbot, L., and Yao, L.S. (1983) Flow in curved pipes. Ann. Rev. Fluid Mech. 15, 461–512.
Boris, J.P. (1989) New directions in computational fluid dynamics. Ann. Rev. Fluid Mech. 21, 345–385.
Bradshaw, P. (1987) Turbulent secondary flows. Ann. Rev. Fluid Mech. 19, 53–74.
Cheng, K.C., and Yuen, F.P. (1987) Flow visualization studies on secondary flow patterns in straight tubes downstream of a 180 deg bend and in isothermally heated horizontal tubes. J. Heat Transfer 109, 49–54.
Cheng, K.C., Inaba, T., and Akiyama, M. (1983) Flow visualisation studies of secondary flow patterns and centrifugal instability in curved circular and semicircular pipes. Proc. 3rd Internat. Symp. on Flow Visualisation, Ann Arbor (ed. W.J. Yang), pp. 531–536.
Choi, U.S., Talbot, L., and Cornet, I. (1979) Experimental study of wall shear rates in the entry region of a curved tube. J. fluid Mech. 93, 465–489.
Chorin, A.J. (1967) A numerical method for solving incompressible viscous flow problems. J. Comput. Phys. 2, 12–26.
Collins, W.M., and Dennis, S.C.R. (1975) The steady motion of a viscous fluid in a curved tube. Quart. J. Mech. Appl. Math. 28, 133–156.
Dean, W.R. (1928) Fluid motion in a curved channel Proc. Roy. Soc. London Ser. A 121, 402–420.
Demuren, A.O., and Rodi, W. (1984) Calculation of turbulence-driven secondary motions in noncircular ducts. J. Fluid Mech. 140, 189–222.
Dennis, S.C.R., and Ng, M. (1982) Dual solutions for steady laminar flow through a curved tube. Quart. J. Mech. Appl. Math. 35, 305–324.
Durrett, R.P., Stevenson, W.H., and Thompson, H.D. (1985) Radial and axial turbulent flow measurements with an LDV in an axisymmetric sudden expansion air flow. Internat. Symp. on Laser Anemometry (eds. A. Dybbs and P.A. Fund), ASME Special Publ. FED-33, 127–133.
Enayet, M.M., Gibson, M.M., Taylor, A.M.K.P., and Yianneskis, M. (1982) Laser-Doppler measurements of laminar and turbulent flow in a pipe bend. Internat. J. Heat Fluid Flow 3, 213–219.
Fairbank, J.A., and So, R.M.C. (1987) Upstream and downstream influence of pipe curvature on the flow through a bend. Internat. J. Heat Fluid Flow 8, 211–217.
Hornung, H.G. (1988) Vorticity generation and secondary flows. Proc. 1st National Fluid Dynamics Congress, July 25–28, 1988, Cincinnati, Ohio, pp. 566–571.
Humphrey, J.A.C., Chang S.M., and Modavi, A. (1982) Developing turbulent flow in 180° bend and downstream tangent of square cross sections. Repart No. LBL-14844, Lawerence Berkeley Laboratory, University of California, Berkeley, CA 94720.
Humphrey, J.A.C., Iacovides, H., and Launder, B.E. (1985) Some numerical experiments on developing laminar flow in circular-sectional bends. J. Fluid Mech. 154, 357–375.
Ito, H. (1969) Laminar flow in curved pipes, Z. Angew. Math. Mech. 49, 653–663.
Lai, Y.G. (1990) Near-wall modelling of complex turbulent flows. Ph.D. Thesis, Department of Mechanical and Aerospace Engineering, Arizona State University, Tempe, Arizona.
Lai, Y.G., So, R.M.C., and Zhang, H.S. (1991) Turbulence-driven secondary flows in a curved pipe. Theoret. Comput. Fluid Dynamics, this issue, pp. 163–180.
Liu, N.S. (1976) Finite-difference solution of the Navier-Stokes equations for incompressible three-dimensional internal flows. Proc. 5th Internat. Conf. on Numerical Methods of Fluid Dynamics, pp. 330–335.
Liu, N.S. (1977) Developing flow in a curved pipe. INSERM—Euromech 92 71, 53–64.
Masliyah, J.H. (1980) On laminar flow in curved semicircular ducts. J. Fluid Mech. 99, 460–479.
Nandakumar, K., and Masliyah, J.H. (1982) Bifurcation in steady laminar flow through curved tubes. J. Fluid Mech. 119, 475–490.
Olson, D.E., and Snyder, B. (1985) The upstream scale of flow development in curved circular pipes. J. Fluid Mech. 150, 139–158.
Patankar, S.V. (1980) Numerical Heat Transfer and Fluid Flow, McGraw-Hill, New York.
Patankar, S.V., and Spalding, D.B. (1972) A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows. Internat. J. Heat Mass Transfer 15, 1787–1806.
Patankar, S.V., Pratap, V.S., and Spalding, D.B. (1974) Prediction of laminar flow and heat transfer in helically coiled pipes. J. Fluid Mech. 62, 539–557.
Patankar, S.V., Pratap, V.S., and Spalding, D.B. (1975) Prediction of turbulent flow in curved pipes. J. Fluid Mech. 67, 583–595.
Rowe, M. (1970) Measurements and computation of flow in pipe bends. J. Fluid Mech. 43, 771–783.
Singh, M.P. (1974) Entry flow in a curve pipe. J. Fluid Mech. 65, 517–539.
Soh, W.Y., and Berger, S.A. (1984) Laminar entrance flow in a curved pipe. J. Fluid Mech. 148, 109–135.
Spalding, D.B. (1972) A novel finite difference formulation for differencing expressions involving both first and second derviatives. Internat. J. Numer. Methods Engrg. 4, 551–559.
Stewartson, K., Cebecci, T., and Chang, K.C. (1980) A boundary-layer collision in a curved duct. Quart. J. Mech. Appl. Math. 33, 59–75.
Talbot, L., and Wong, S.J. (1982) A note on boundary-layer collision in a curved pipe. J. Fluid Mech. 122, 505–510.
van Dyke, M. (1978) Extended Stokes series: laminar flow through a loosely coiled pipe. J. Fluid Mech. 86, 129–145.
Yao, L.S., and Berger, S.A. (1975) Entry flow in a curved pipe. J. Fluid Mech. 67, 177–196.
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Communicated by M.Y. Hussaini
This work was supported by the Office of Naval Research under Grant No. N0014-81-K-0428 and by DTRC, Annapolis, Maryland, under Contract No. N00167-86-K-0075. Also, support in the form of an IPA awarded to RMCS during his sabbatical leave at DTRC, Annapolis, Maryland, in the spring of 1990 is gratefully acknowledged.
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So, R.M.C., Zhang, H.S. & Lai, Y.G. Secondary cells and separation in developing laminar curved-pipe flows. Theoret. Comput. Fluid Dynamics 3, 141–162 (1991). https://doi.org/10.1007/BF00271799
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DOI: https://doi.org/10.1007/BF00271799