An experimental and numerical study of the Dean problem: flow development towards two-dimensional multiple solutions
An experimental and numerical study investigating the flow development and fully developed flows of an incompressible Newtonian fluid in a curved duct of square cross section with a curvature ratio of 15.1 is presented. Numerical simulations of flow development from a specified inlet profile were performed using a parabolized form of the steady three-dimensional Navier-Stokes equations. No symmetry conditions were imposed. In general there was good agreement between the numerical predictions of the developing axial velocity profiles and LDV measurements. In addition, for computational expediency, the two-dimensional solution structure was calculated by imposing fully developed conditions together with symmetry conditions along the horizontal duct centreline.
Laser-Doppler measurements of axial velocity and flow visualization at Dean number Dn = 125, 137 and 150, revealed a steady and symmetric two-vortex flow at Dn = 125, and a steady and symmetric four-vortex flow at both Dn = 137 and 150 (Dn = Re/(R/a)½, where Re is the Reynolds number, R is the radius of curvature of the duct and a is the duct dimension). Axial velocity measurements showed that the four-vortex flow at Dn = 150 developed to the solution predicted by the two-dimensional numerical simulation. However, the four-vortex flow at Dn = 137 was still developing when the flow had reached the end of the 240° axial length of the duct. A numerical investigation for Dean numbers in the range of 50 to 175 revealed that at the limit point of the two-cell to four-cell transition the development length appeared to be infinite, and thereafter decreased for increasing Dean numbers. The behaviour of decreasing development length of the four-vortex flow with increasing Dean number has not been reported previously.
Using a symmetrically positioned pin at θ = 5° to induce the four-cell flows, the two-dimensional solution structure for Dn [less-than-or-eq, slant] 150 was experimentally observed for the first time. Experiments were consistent with the prediction by Winters (1987) that four-vortex flows are stable to symmetric perturbations, but unstable to asymmetric perturbations. Experimental and numerical investigations suggested that, when perturbed asymmetrically, the four-vortex flow might evolve to flows with sustained spatial oscillations farther downstream.(Published Online April 26 2006)
(Received August 16 1991)
(Revised April 13 1992)