The development and structure of primary and secondary flow in a curved square duct

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Abstract
The development of laminar flow in a 180° section of a curved square duct (R/d = 6.45) was studied by laser-Doppler anemometry (LDA). The streamwise flow-velocity component Vϕ and the secondary flow component Vr were measured as a function of Dean number and of the azimuthal angle ϕ. The development of the streamwise flow component was found to be connected with a strong momentum transfer towards the outer wall between ϕ = 0° and ϕ = 60°, with a partial back-transfer of the momentum towards the duct centre (ϕ = 45°–108°), and with little further change of the momentum between ϕ = 108° and 180° near the outer wall. The measurements of the Vr component showed at least one vortex pair in the secondary flow. A second vortex pair with opposing sense of circulation was found to develop near the outer wall only for Dean numbers between 150 and 300, in agreement with numerical calculations. This second vortex pair was found in the region between ϕ = 108 and 171°. Between ϕ = 60 and 108° it was not possible to identify the second vortex pair in the developing flow. However, developing laminar-flow numerical calculations by Humphrey (1982) show that it also arises for K = 485 in a 180° square duct with R/d = 3.35 and Re = 890, and it is a function of inlet flow conditions. From the measured stream-function maximum of the second vortex pair it may be deduced that a curved duct with 180° (20 hydraulic diameters) bend angle is not sufficient to reach fully developed flow conditions at d/R = 1/6.45 and K = 226.