Hostname: page-component-7479d7b7d-767nl Total loading time: 0 Render date: 2024-07-10T19:24:22.743Z Has data issue: false hasContentIssue false
  • English
  • Français

Some numerical experiments on developing laminar flow in circular-sectioned bends

Published online by Cambridge University Press:  20 April 2006

J. A. C. Humphrey ,
H. Iacovides  and
B. E. Launder
J. A. C. Humphrey
Affiliation:
University of California, Berkeley, California
H. Iacovides
Affiliation:
UMIST, Manchester
B. E. Launder
Affiliation:
UMIST, Manchester
Get access Rights & Permissions [Opens in a new window]

Abstract

The paper reports numerical solutions to a semi-elliptic truncation of the Navier–Stokes equations for the case of developing laminar flow in circular-sectioned bends over a range of Dean numbers. The ratios of bend radius to pipe radius are 7:1 and 20:1, corresponding with the configurations examined experimentally by Talbot and his co-workers in recent years. The semi-elliptic treatment facilitates a much finer grid than has been possible in earlier studies. Numerical accuracy has been further improved by assuming radial equilibrium over a thin sublayer immediately adjacent to the wall and by re-formulating the boundary conditions at the pipe centre.

Streamwise velocity profiles at Dean numbers of 183 and 565 are in excellent agreement with laser-Doppler measurements by Agrawal, Talbot & Gong (1978). Good, albeit less complete, accord is found with the secondary velocities, though the differences that exist may be mainly due to the difficulty of making these measurements. The paper provides new information on the behaviour of the streamwise shear stress around the inner line of symmetry. Upstream of the point of minimum shear stress, our numerical predictions display a progressive shift towards the result of Stewartson, Cebici & Chang (1980) as the Dean number is successively raised. Downstream of the minimum, however, in contrast with the monotonic approach to an asymptotic level reported by Stewartson, the numerical solutions display a damped oscillatory behaviour reminiscent of those from Hawthorne's (1951) inviscid-flow calculations. The amplitude of the oscillation grows as the Dean number is raised.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 154 , May 1985 , pp. 357 - 375
Copyright
© 1985 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

Agrawal, Y., Talbot, L. & Gong, K. 1978 Laser anemometer study of flow development in curved pipes. J. Fluid Mech. 85, 497–518.Google Scholar
Azzola, J. & Humphrey, J. A. C. 1984 Developing turbulent flow in a 180 curved pipe and its downstream tangent. In 2nd Int. Symp. on Applications of Laser Anemometry to Fluid Mechanics, Lisbon.
Barua, S. N. 1963 On secondary flow in stationary curved pipes. Q. J. Mech. Appl. Maths 16, 61–77.Google Scholar
Berger, S. A., Talbot, L. & Yao, L.-S. 1983 Flow in curved pipes. Ann. Rev. Fluid Mech. 15, 461–512.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, 465–489.Google Scholar
Collins, W. & Dennis, S. C. R. 1975 The steady motion of a viscous fluid in a curved tube. Q. J. Mech. Appl. Maths 28, 133–156.Google Scholar
Cuming, H. G. 1952 The secondary flow in curved pipes. Aero. Res. Counc. Rep. Mem. No. 2880.
Han, T. Y., Humphrey, J. A. C. & Launder, B. E. 1981 A comparison of hybrid and quadratic-upstream differencing in high Reynolds number elliptic flows. Comp. Meth. Appl. Mech. Engng 29, 81.Google Scholar
Hawthorne, W. R. 1951 Secondary circulation in fluid flow. Proc. R. Soc. Lond. A 206, 374–387.Google Scholar
Huang, P. G., Launder, B. E. & Leschziner, M. A. 1985 Discretization of non-linear convection processes: A broad range comparison of four schemes. To appear in Comp. Meth. Appl. Mech. Engng.
Humphrey, J. A. C. 1977 Numerical calculation of developing laminar flow in pipes of arbitrary curvature radius. Can. J. Chem. Engng 56, 151–164.Google Scholar
Iacovides, H. & Launder, B. E. 1984 PSL - An economical approach to the numerical analysis of near-wall elliptic flow. Trans. ASME I: J. Fluid Engng 106, 245Google Scholar
Ito, H. 1969 Laminar flow in curved pipes. Z. angew. Math. Mech. 49, 653–663.Google Scholar
Johnston, J. P. 1978 Internal flows. In Turbulence (ed. P. Bradshaw). Topics in Applied Physics, vol. 12, chap. 3. Springer.
Leonard, B. P. 1979 A stable and accurate convective modelling procedure based on quadratic upstream interpolation. Comp. Meth. Appl. Mech. Engng 19, 59.Google Scholar
Levy, R., Briley, W. R. & McDonald, H. 1983 Viscous primary/secondary flow analysis for use with non-orthogonal coordinate systems. AIAA Paper 83-056.
Liu, N.-S. 1976 Finite-difference solution of the Navier-Stokes equations for three-dimensional internal flow. In Proc. 5th Intl Conf. on Num. Meth. in Fluid Dynamics, pp. 330–335.
Liu, N.-S. 1977 Developing flow in a curved pipe. INSERM-Euromech 92 71, 53–64.Google Scholar
Patankar, S. V. 1980 Numerical Heat Transfer and Fluid Flow. Hemisphere Publishing Corp./McGraw-Hill.
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, 539–551.Google Scholar
Patankar, S. V. & Spalding, D. B. 1972 A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows. Int J. Heat Mass Transfer 15, 1787.Google Scholar
Pratap, V. S. & Spalding, D. B. 1975 Numerical computations of the flow in curved ducts. Aero. Quart. 26, 219–228.Google Scholar
Rushmore, W. L. 1975 Theoretical investigation of curved pipe flows. Ph.D. thesis, New York State University at Buffalo, 160 pp.
Singh, M. P. 1974 Entry flow in a curved pipe. J. Fluid Mech. 65, 517–539.Google Scholar
Soh, W. Y. 1983 Laminar entrance flow in a curved pipe. Ph.D. thesis, University of California, Berkeley, 91 pp.
Soh, W. Y. & Berger, S. A. 1984 Laminar entrance flow in a curved pipe. J. Fluid Mech. 148, 109–135.Google Scholar
Stewartson, K., Cebeci, T. & Chang, K. C. 1980 A boundary-layer collision in a curved duct. Q. J. Mech. Appl. Maths 33, 59–75.Google Scholar
Talbot, L. & Wong, S. J. 1982 A note on boundary-layer collision in a curved pipe. J. Fluid Mech. 122, 505–510.Google Scholar
Yao, L-S. & Berger, S. A. 1975 Entry flow in a curved pipe. J. Fluid Mech. 67, 177–196.Google Scholar