Skip to main content Accessibility help

Reynolds number dependence of mean flow structure in square duct turbulence



We have performed direct numerical simulations of turbulent flows in a square duct considering a range of Reynolds numbers spanning from a marginal state up to fully developed turbulent states at low Reynolds numbers. The main motivation stems from the relatively poor knowledge about the basic physical mechanisms that are responsible for one of the most outstanding features of this class of turbulent flows: Prandtl's secondary motion of the second kind. In particular, the focus is upon the role of flow structures in its generation and characterization when increasing the Reynolds number. We present a two-fold scenario. On the one hand, buffer layer structures determine the distribution of mean streamwise vorticity. On the other hand, the shape and the quantitative character of the mean secondary flow, defined through the mean cross-stream function, are influenced by motions taking place at larger scales. It is shown that high velocity streaks are preferentially located in the corner region (e.g. less than 50 wall units apart from a sidewall), flanked by low velocity ones. These locations are determined by the positioning of quasi-streamwise vortices with a preferential sign of rotation in agreement with the above described velocity streaks' positions. This preferential arrangement of the classical buffer layer structures determines the pattern of the mean streamwise vorticity that approaches the corners with increasing Reynolds number. On the other hand, the centre of the mean secondary flow, defined as the position of the extrema of the mean cross-stream function (computed using the mean streamwise vorticity), remains at a constant location departing from the mean streamwise vorticity field for larger Reynolds numbers, i.e. it scales in outer units. This paper also presents a detailed validation of the numerical technique including a comparison of the numerical results with data obtained from a companion experiment.


Corresponding author

Email address for correspondence:


Hide All
Biau, D., Soueid, H. & Bottaro, A. 2008 Transition to turbulence in duct flow. J. Fluid Mech. 596, 133142.
Brundrett, E. & Baines, W. 1964 The production and diffusion of vorticity in duct flow. J. Fluid Mech. 19, 375394.
Gavrilakis, S. 1992 Numerical simulation of low-Reynolds-number turbulent flow through a straight square duct. J. Fluid Mech. 244, 101129.
Gessner, F. 1973 The origin of secondary flow in turbulent flow along a corner. J. Fluid Mech. 58, 125.
Haldenwang, P., Labrosse, G., Abboudi, S. & Deville, M. 1984 Chebyshev 3-d spectral and 2-d pseudo-spectral solvers for the Helmholtz equation. J. Comput. Phys. 55, 115128.
Huser, A. & Biringen, S. 1993 Direct numerical simulation of turbulent flow in a square duct. J. Fluid Mech. 257, 6595.
Jiménez, J. & Moin, P. 1991 The minimal flow unit in near-wall turbulence. J. Fluid Mech. 225, 213240.
Jones, O. 1976 An improvement in the calculation of turbulent friction in rectangular ducts. ASME J. Fluids Engng 98, 173181.
Kawahara, G., Ayukawa, K., Ochi, J., Ono, F. & Kamada, E. 2000 Wall shear stress and Reynolds stresses in a low Reynolds number turbulent square duct flow. Trans. JSME B 66 (641), 95102.
Kida, S. & Miura, H. 1998 Swirl condition in low-pressure vortices. J. Phys. Soc. Japan 67 (7), 21662169.
Kim, J., Moin, P. & Moser, R. D. 1987 Turbulence statistics in fully developed channel flow at low Reynolds number. J. Fluid Mech. 177, 133166.
Madabhushi, R. K. & Vanka, S. P. 1991 Large eddy simulation of turbulence-driven secondary flow in a square duct. Phys. Fluids A 3 (11), 27342745.
Melling, A. & Whitelaw, J. 1976 Turbulent flow in a rectangular duct. J. Fluid Mech. 78, 289315.
Nikuradse, J. 1926 Untersuchungen über die Geschwindigkeitsverteilung in turbulenten Strömungen. PhD Thesis, Göttingen. VDI Forsch. 281.
Prandtl, L. 1926 Über die ausgebildete turbulenz. Verh. 2nd Intl Kong. Fur Tech. Mech., Zurich [English transl. NACA Tech. Memo. 435].
Uhlmann, M., Pinelli, A., Kawahara, G. & Sekimoto, A. 2007 Marginally turbulent flow in a square duct. J. Fluid Mech. 588, 153162.
Verzicco, R. & Orlandi, P. 1996 A finite-difference scheme for three-dimensional incompressible flows in cylindrical coordinates. J. Comput. Phys. 123, 402414.
MathJax is a JavaScript display engine for mathematics. For more information see


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