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Laminar entrance flow in a curved pipe

Published online by Cambridge University Press:  20 April 2006

W. Y. Soh  and
S. A. Berger
W. Y. Soh
Affiliation:
Department of Mechanical Engineering, University of California, Berkeley, California 94720
S. A. Berger
Affiliation:
Department of Mechanical Engineering, University of California, Berkeley, California 94720
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Abstract

The full elliptic Navier–Stokes equations have been solved for entrance flow into a curved pipe using the artificial compressibility technique developed by Chorin (1967). The problem is formulated for arbitrary values of the curvature ratio and the Dean number. Calculations are carried out for two curvature ratios, a/R = 1/7 and 1/20, and for Dean number ranging from 108.2 to 680.3, in a computational mesh extending from the inlet immediately adjacent to the reservoir to the fully developed downstream region.

Secondary flow separation near the inner wall is observed in the developing region of the curved pipe. The separation and the magnitude of the secondary flow are found to be greatly influenced by the curvature ratio. As observed in the experiments of Agrawal, Talbot & Gong (1978) we find: (i) two-step plateau-like axial-velocity profiles for high Dean number, due to the secondary flow separation, and (ii) doubly peaked axial-velocity profiles along the lines parallel to the plane of symmetry, due to the highly distorted secondary-flow vortex structure.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 148 , November 1984 , pp. 109 - 135
Copyright
© 1984 Cambridge University Press

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References

Adler, V. M. 1934 Strömung in gekrümmten Rohren. Z. angew. Math. Mech. 14, 257.Google Scholar
Agrawal, Y., Talbot, L. & Gong, K. 1978 Laser anemometer study of flow development in curved circular pipes. J. Fluid Mech. 85, 497.Google Scholar
Akiyama, M. & Cheng, K. C. 1971 Boundary vorticity method for laminar forced convection heat transfer in curved pipes. Intl J. Heat Mass Transfer 14, 1659.Google Scholar
Allen, J. S. & Cheng, S. I. 1970 Numerical solutions of the compressible Navier—Stokes equations for the laminar near wake. Phys. Fluids 13, 37.Google Scholar
Austin, L. 1971 The development of viscous flow within helical coils. Ph.D. thesis, University of Utah, Salt Lake City.
Austin, L. R. & Seader, J. D. 1973 Fully developed viscous flow in coiled circular pipes. AIChE J. 19, 85.Google Scholar
Barua, S. N. 1963 On secondary flow in stationary curved pipes. Q. J. Mech. Appl. Maths 16, 61.Google Scholar
Berger, S. A., Talbot, L. & Yao, L.-S. 1983 Flow in curved pipes. Ann. Rev. Fluid Mech. 15, 461.Google Scholar
Caro, C. G., Fitz-Gerald, J. M. & Schroter, R. C. 1971 Atheroma and arterial wall shear: observation, correlation and proposal of a shear dependent mass transfer mechanism for atherogenesis. Proc. R. Soc. Lond. B 177, 109.Google Scholar
Choi, U. S., Talbot, L. & Cornet, I. 1979 Experimental study of wall shear rates in the entry region of a curved tube. J. Fluid Mech. 93, 465.Google Scholar
Chorin, A. J. 1967 A numerical method for solving incompressible viscous flow problems. J. Comp. Phys. 2, 12.Google Scholar
Chorin, A. J. 1968 Numerical solution of the Navier—Stokes equations. Math. Comp. 22, 745.Google Scholar
Collins, W. M. & Dennis, S. C. R. 1975 The steady motion of a viscous fluid in a curved tube. Q. J. Mech. Appl. Maths 28, 133.Google Scholar
Dean, W. R. 1927 Note on the motion of fluid in a curved pipe. Phil. Mag. 20, 208.Google Scholar
Dean, W. R. 1928 The streamline motion of fluid in a curved pipe. Phil. Mag. 30, 673.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.Google Scholar
Dennis, S. C. R. & Ng, M. 1982 Dual solutions for steady laminar flow through a curved tube. Q. J. Mech. Appl. Maths 35, 305.Google Scholar
Eustice, J. 1910 Flow of water in curved pipes. Proc. R. Soc. Lond. A 84, 107.Google Scholar
Eustice, J. 1911 Experiments of streamline motion in curved pipes. Proc. R. Soc. Lond. A 85, 119.Google Scholar
Fortin, M., Peyret, R. & Teman, R. 1971 Résolution numérique des équations de Navier—Stokes pour un fluide incompressible. J. Méc. 10, 357.Google Scholar
Fry, D. L. 1968 Acute vascular endothelial changes associated with increased blood velocity gradients. Circ. Res. 22, 165.Google Scholar
Fry, D. L. 1973 Atherogenesis: initiating factors. Ciba Symp. 12, 93.Google Scholar
Grabowski, W. J. & Berger, S. A. 1976 Solutions of the Navier—Stokes equations for vortex breakdown. J. Fluid Mech. 75, 525.Google Scholar
Greenspan, D. 1973 Secondary flow in a curved tube. J. Fluid Mech. 57, 167.Google Scholar
Harlow, F. H. & Welch, J. E. 1965 Numerical calculation of time dependent viscous incompressible flow of fluid with free surface. Phys. Fluids 8, 2182.Google Scholar
Hasson, D. 1955 Streamline flow resistance in coils. Res. Corresp. 1: S1.Google Scholar
Hirt, C. W. & Cook, J. L. 1972 Calculating three-dimensional flow around structures and over rough terrain. J. Comp. Phys. 10, 324.Google Scholar
Humphrey, J. A. C. 1977 Flow in ducts with curvature and roughness. Ph.D. thesis, Imperial College of Science and Technology.
Itó, H. 1969 Laminar flow in a curved pipe. Z. angew. Math. Mech. 49, 653.Google Scholar
Liu, N.-S. 1977 Developing flow in a curved pipe. INSERM—Euromech 92 71, 53.Google Scholar
Masliyah, J. H. 1980 On laminar flow in curved semicircular ducts. J. Fluid Mech. 99, 469.Google Scholar
McConalogue, D. J. & Srivastava, R. S. 1968 Motion of fluid in a curved tube. Proc. R. Soc. Lond. A 307, 37.Google Scholar
Patankar, S. V., Pratap, V. S. & Spalding, D. B. 1974 Prediction of laminar flow and heat transfer in helically coiled pipes. J. Fluid Mech. 62, 539.Google Scholar
Pedley, T. J. 1980 The Fluid Mechanics of Large Blood Vessels, chap. 4. Cambridge University Press.
Peskin, C. S. 1972 Flow patterns around heart valves: a numerical method. J. Comp. Physics 10, 252.Google Scholar
Plows, W. H. 1968 Some numerical results for two-dimensional steady laminar Bénard convection. Phys. Fluids 11, 1593.Google Scholar
Singh, M. P. 1974 Entry flow in a curved pipe. J. Fluid Mech. 65, 517.Google Scholar
Smith, F. T. 1976 Fluid flow into a curved pipe. Proc. R. Soc. Lond. A 351, 71.Google Scholar
Soh, W. Y. 1983 Laminar entrance flow in a curved pipe. Ph.D. thesis, University of California, Berkeley.
Stewartson, K., Cebeci, T. & Chang, K. C. 1980 A boundary-layer collision in a curved duct. Q. J. Mech. Appl. Maths 33, 59.Google Scholar
Stewartson, K. & Simpson, C. J. 1982 On a singularity initiating a boundary-layer collision. Q. J. Mech. Appl. Maths 35, 1.Google Scholar
Talbot, L. & Wong, S. J. 1982 A note on boundary-layer collision in a curved pipe. J. Fluid Mech. 122, 505.Google Scholar
Taylor, G. I. 1929 The criterion for turbulence in curved pipes. Proc. R. Soc. Lond. A 124, 243.Google Scholar
Thompson, J. 1876 On the origin of winding of rivers in alluvial plains, with remarks on the flow of water round bends in pipes. Proc. R. Soc. Lond. A 25, 5.Google Scholar
Truesdell, L. C. & Adler, R. J. 1970 Numerical treatment of fully developed laminar flow in helically coiled tubes. AIChE J. 16, 1010.Google Scholar
Van Dyke, M. 1978 Extended Stokes series: laminar flow through a loosely coiled pipe. J. Fluid Mech. 86, 129.Google Scholar
Viecelli, J. A. 1971 A computing method for incompressible flows bounded by moving walls. J. Comp. Phys. 8, 119.Google Scholar
White, C. M. 1929 Streamline flow through curved pipes. Proc. R. Soc. Lond. A 123, 645.Google Scholar
Williams, G. P. 1969 Numerical integration of the three-dimensional Navier—Stokes equations for incompressible flow. J. Fluid Mech. 37, 727.Google Scholar
Williams, G. S., Hubbell, C. W. & Fenkell, G. H. 1902 Experiments at Detroit, Mich., on the effect of curvature upon the flow of water in pipes. Trans. ASCE 47, 1.Google Scholar
Yao, L.-S. & Berger, S. A. 1975 Entry flow in a curved pipe. J. Fluid Mech. 88, 339.Google Scholar