Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-19T15:26:37.387Z Has data issue: false hasContentIssue false
  • English
  • Français

The development of the turbulent flow in a bent pipe

Published online by Cambridge University Press:  26 April 2007

PHILLIP L. WILSON  and
FRANK T. SMITH
PHILLIP L. WILSON
Affiliation:
Department of Mathematics & Statistics, University of Canterbury, Private Bag 4800, Christchurch 8140, New Zealand
FRANK T. SMITH
Affiliation:
Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

The three-dimensional incompressible turbulent flow through a slender bent pipe of simple cross-section is analysed, the pipe gradually bending the rapid flow through a substantial angle. The ratio of the relative radius of curvature to the magnitude of the turbulent fluctuations is crucial: analysis of the entry region involving exact solutions of the governing equations shows three different downstream developments, depending on the magnitude of that ratio. The main velocity components are found in each case, and one downstream development studied in detail is when turbulence dominates the flow.

The main novel points and results are as follows. (i) The present physical situation which arises commonly in industrial settings has been little studied previously by theory or experiments. (ii) The working applies for any two-tier mixing-length model. (iii) As a most surprising feature, the fully developed flow far downstream is not unique, being found to depend instead on the global flow behaviour (thus the centreline velocity is not determined simply by the pressure drop, in contrast to the laminar case). (iv) A quite accurate predictive tool based on approximation is suggested for the downstream flow. (v) Crossflow maxima are found to occur very close to the walls, as observed in experiments. (vi) Other comparisons are made with experimental data and prove generally favourable.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 578 , 10 May 2007 , pp. 467 - 494
Copyright
Copyright © Cambridge University Press 2007

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

REFRENCES

Barbin, A. R. & Jones, J. B. 1963 Turbulent flow in the inlet region of a smooth pipe. Trans. ASME D: J. Basic Engng 85, 29–34.CrossRefGoogle Scholar
Bush, W. B. & Fendell, F. E. 1972 Asymptotic analysis of turbulent channel and boundary-layer flow. J. Fluid Mech. 56, 657–668.CrossRefGoogle Scholar
Cebeci, T. & Smith, A. M. O. 1974 Analysis of Turbulent Boundary Layers. Academic.Google Scholar
Chinni, R., Sahai, V. & Munukutla, S. 1996 Computational study of developing turbulent pipe flow. In Hydrodynamics (ed. LeungChwang, Lee Chwang, Lee). Balkema.Google Scholar
Degani, A. T., Smith, F. T. & Walker, J. D. A. 1993 The structure of a three-dimensional turbulent boundary layer. J. Fluid Mech. 250, 43–68.CrossRefGoogle Scholar
Ellis, L. B. & Joubert, P. N. 1974 Turbulent shear flow in a curved duct. J. Fluid Mech. 62, 65–84.CrossRefGoogle Scholar
Huffman, G. D. & Bradshaw, P. 1972 A note on von Kármán's constant in low Reynolds number turbulent flow. J. Fluid Mech. 53, 45–60.CrossRefGoogle Scholar
Hunt, I. A. & Joubert, P. N. 1979 Effects of small streamline curvature on turbulent duct flow. J. Fluid Mech. 91, 633–659.CrossRefGoogle Scholar
Klebanoff, P. S. 1954 Characteristics of turbulence in a boundary layer with zero pressure gradient. NACA TN 3178.Google Scholar
Knight, D. D. 1998 Inviscid compressible flow. In The Handbook of Fluid Dynamics (ed. Johnson, R. W.). CRC.Google Scholar
Laufer, J. 1949 Investigation of turbulent flow in a two-dimensional channel. NACA Rep. 1053.Google Scholar
Laufer, J. 1952 The structure of turbulence in fully developed pipe flow. NACA Rep. 1174.Google Scholar
Mager, A. 1964 Three-dimensional laminar boundary layers. In Theory of Laminar Flows (ed. Moore, F. K.), High Speed Aerodynamics and Jet Propulsion, vol. 4. Oxford University Press.Google Scholar
Melling, A. & Whitelaw, J. H. 1976 Turbulent flow in a rectangular duct. J. Fluid Mech. 78, 289–315.CrossRefGoogle Scholar
Mellor, G. L. 1972 The large Reynolds number, asymptotic theory of turbulent boundary layers. Intl J. Engng Sci. 10, 851–873.CrossRefGoogle Scholar
Nakao, S. 1986 Turbulent flow in square ducts after an expansion. AIAA J. 24 (6), 979–982.CrossRefGoogle Scholar
Neish, A. & Smith, F. T. 1988 The turbulent boundary layer and wake of an aligned flat plate. J. Engng Maths 22, 15–42.CrossRefGoogle Scholar
Nikuradse, J. 1933 Law of flow in rough pipes. NACA TM 1292.Google Scholar
Ockendon, H., Ockendon, J. R. & Falle, S. A. E. G. 2001 The turbulent boundary layer and wake of an aligned flat plate. J. Fluid Mech. 445, 187–206.CrossRefGoogle Scholar
Schlichting, H. & Gersten, K. 2000 Boundary Layer Theory, 8th edn. Springer.CrossRefGoogle Scholar
Schwarz, W. R. & Bradshaw, P. 1994 Turbulence structural changes for a three-dimensional turbulent boundary layer in a 30° bend. J. Fluid Mech. 272, 183–209.CrossRefGoogle Scholar
Smith, F. T. & Li, L. 2002 Swirl flow effects in a duct bending through a substantial angle. J. Engng Maths 43, 315–346.CrossRefGoogle Scholar
Sychev, V. V. 1987 On turbulent boundary layer separation. In Boundary Layer Separation (ed. Smith, F. T. & Brown, S. N.). Springer.Google Scholar
Talbot, L. & Wong, S. J. 1982 A note on boundary-layer collision in a curved pipe. J. Fluid Mech. 122, 505–510.CrossRefGoogle Scholar
Wilson, P. L. 2003 On the core flow and turbulent boundary layer in a curved duct. PhD thesis, University of London.Google Scholar
Wilson, P. L. & Smith, F. T. 2005 a Compressible swirling flow through a significantly bent duct. Comput. Fluids 34, 891–926.CrossRefGoogle Scholar
Wilson, P. L. & Smith, F. T. 2005 b A three-dimensional pipe flow adjusts smoothly to the sudden onset of a bend. Phys. Fluids 17, 048102.CrossRefGoogle Scholar
Zhang, J. & Lakshminarayana, B. 1990 Computation and turbulence modeling for three-dimensional boundary layers including turbomachinery flows. AIAA J. 1861–1869.CrossRefGoogle Scholar