Skip to main content Accessibility help

Downstream decay of fully developed Dean flow

  • Jesse T. Ault (a1), Kevin K. Chen (a1) and Howard A. Stone (a1)


Direct numerical simulations were used to investigate the downstream decay of fully developed flow in a $180^{\circ }$ curved pipe that exits into a straight outlet. The flow is studied for a range of Reynolds numbers and pipe-to-curvature radius ratios. Velocity, pressure and vorticity fields are calculated to visualize the downstream decay process. Transition ‘decay’ lengths are calculated using the norm of the velocity perturbation from the Hagen–Poiseuille velocity profile, the wall-averaged shear stress, the integral of the magnitude of the vorticity, and the maximum value of the $Q$ -criterion on a cross-section. Transition lengths to the fully developed Poiseuille distribution are found to have a linear dependence on the Reynolds number with no noticeable dependence on the pipe-to-curvature radius ratio, despite the flow’s dependence on both parameters. This linear dependence of Reynolds number on the transition length is explained by linearizing the Navier–Stokes equations about the Poiseuille flow, using the form of the fully developed Dean flow as an initial condition, and using appropriate scaling arguments. We extend our results by comparing this flow recovery downstream of a curved pipe to the flow recovery in the downstream outlets of a T-junction flow. Specifically, we compare the transition lengths between these flows and document how the transition lengths depend on the Reynolds number.


Corresponding author

Email address for correspondence:


Hide All

Present address: Department of Aerospace and Mechanical Engineering, USC, Los Angeles, CA 90089, USA.



Hide All
Anwer, M. & So, R. M. C. 1993 Swirling turbulent flow through a curved pipe. Part I: effect of swirl and bend curvature. Exp. Fluids 14, 8596.
Anwer, M., So, R. M. C. & Lai, Y. G. 1989 Perturbation by and recovery from bend curvature of a fully developed turbulent pipe flow. Phys. Fluids 1, 13871397.
Atkinson, B., Brocklebank, M. P., Card, C. C. H. & Smith, J. M. 1969 Low Reynolds number developing flows. AIChE J. 15, 548553.
Austin, L. R. & Seader, J. D. 1973 Fully developed viscous flow in coiled circular pipes. AIChE J. 19, 8594.
Berger, S. A. & Talbot, L. 1983 Flow in curved pipes. Annu. Rev. Fluid Mech. 15, 461512.
Chen, K. K., Rowley, C. W. & Stone, H. A. 2015 Vortex dynamics in a pipe T-junction: recirculation and sensitivity. Phys. Fluids 27 (3), 034107.
Dean, W. R. 1927 Note on the motion of fluid in a curved pipe. Phil. Mag. 20, 208223.
Dean, W. R. 1928 The stream-line motion of fluid in a curved pipe. Phil. Mag. 5, 673695.
Dennis, S. C. R. & Riley, N. 1991 On the fully developed flow in a curved pipe at large Dean number. Proc. R. Soc. Lond. A 434, 473478.
Enayet, M. M., Gibson, M. M., Taylor, A. M. K. P. & Yianneskis, M. 1982 Laser-doppler measurements of laminar and turbulent flow in a pipe bend. Intl J. Heat Fluid Flow 3, 213219.
Fairbank, J. A. & So, R. M. C. 1987 Upstream and downstream influence of pipe curvature on the flow through a bend. Intl J. Heat Fluid Flow 8, 211217.
Fox, R. W., Pritchard, P. J. & McDonald, A. T. 2009 Introduction to Fluid Mechanics, 7th edn. John Wiley & Sons.
Hellström, F. & Fuchs, L.2007 Numerical computations of steady and unsteady flow in bended pipes. In 37th AIAA Fluid Dynamics Conference and Exhibit, 25–28 June 2007, Miami, FL.
Hellström, L. H. O., Zlatinov, M. B., Cao, G. & Smits, A. J. 2013 Turbulent pipe flow downstream of a $90^{\circ }$  bend. J. Fluid Mech. 735, R7(1–12).
Hunt, J. C. R., Wray, A. & Moin, P.1988 Eddies, stream, and convergence zones in turbulent flows. Tech. Rep. CTR-S88. Center for Turbulence Research.
Issa, R. I. 1985 Solution of the implicitly discretised fluid flow equations by operator-splitting. J. Comput. Phys. 62, 4065.
Issa, R. I. 1986 The computation of compressible and incompressible recirculating flows by a non-iterative implicit scheme. J. Comput. Phys. 62, 6682.
Kalpakli, A., Örlü, R., Tillmark, N. & Alfredsson, P. H. 2011 Pulsatile turbulent flow through pipe bends at high Dean and Womersley numbers. J. Phys. 318, 092023.
Liu, S. & Masliyah, J. H. 1996 Steady developing laminar flow in helical pipes with finite pitch. Intl J. Comput. Fluid Dyn. 6, 209224.
Mohanty, A. K. & Asthana, S. B. L. 1978 Laminar flow in the entrance region of a smooth pipe. J. Fluid Mech. 90, 433447.
Olson, D. E. & Snyder, B. 1985 The upstream scale of flow development in curved circular pipes. J. Fluid Mech. 150, 139158.
Pruvost, J., Legrand, J. & Legentilhomme, P. 2004 Numerical investigation of bend and torus flows. Part I: effect of swirl motion on flow structure in U-bend. Chem. Engng Sci. 59, 33453357.
Sakakibara, J. & Machida, N. 2012 Measurement of turbulent flow upstream and downstream of a circular pipe bend. Phys. Fluids 24, 041702.
Singh, M. P. 1974 Entry flow in a curved pipe. J. Fluid Mech. 65, 517539.
Smith, F. T. 1976 Fluid flow into a curved pipe. Proc. R. Soc. Lond. A 351, 7187.
Smits, A. J., Young, S. T. B. & Bradshaw, P. 1979 The effect of short regions of high surface curvature on turbulent boundary layers. J. Fluid Mech. 94, 209242.
So, R. M. C. & Anwer, M. 1993 Swirling turbulent flow through a curved pipe. Part II: recovery from swirl and bend curvature. Exp. Fluids 14, 169177.
Sudo, K., Sumida, M. & Hibara, H. 2000 Experimental investigation on turbulent flow through a circular-sectioned $180^{\circ }$ bend. Exp. Fluids 28, 5157.
Tiwari, P., Antal, S. P. & Podowski, M. l.  Z. 2006 Three-dimensional fluid mechanics of particulate two-phase flows in U-bend and helical conduits. Phys. Fluids 18, 043304.
Tunstall, M. J. & Harvey, J. K. 1968 On the effect of a sharp bend in a fully developed turbulent pipe-flow. J. Fluid Mech. 34, 595608.
Vigolo, D., Radl, S. & Stone, H. A. 2014 Unexpected trapping of particles at a T-junction. Proc. Natl Acad. Sci. USA 111, 47704775.
Weller, H. G., Tabor, G., Jasak, H. & Fureby, C. 1998 A tensorial approach to continuum mechanics using object-oriented techniques. Comput. Phys. 12, 620631.
Winters, K. H. 1987 A bifurcation study of laminar flow in a curved tube of rectangular cross-section. J. Fluid Mech. 180, 343369.
Yao, L. S. & Berger, S. A. 1975 Entry flow in a curved pipe. J. Fluid Mech. 67, 177196.
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed