Oscillatory flows in coiled square ducts

The development of complex velocity fields in curved ducts from an initially parabolic profile is studied using a three-dimensional numerical model of the parabolized Navier–Stokes equations. The velocity profiles are influenced strongly by a geometrical parameter Rc (the radius of curvature) and a dynamic parameter Dn (Dean number, Re/(Rc)1/2). For Rc < 10 and Dn up to 200, the velocity fields develop into the previously observed two- and four-cell solutions that are axially invariant and symmetric about the midplane. For Rc =100 and Dn>125 oscillatory solutions develop which are periodic in the axial direction, but are asymmetric about the midplane. Increasing the Dean number over a narrow range results in a significant increase in the frequency of such oscillations. Grid sensitivity tests indicate that such oscillations are not a numerical artifact. Development of oscillatory solutions is delayed with decreasing radius of curvature. Thus for Rc =10, axially invariant two-dimensional solutions that retain the symmetry about the midplane could be obtained for Dn as high as 300. This trend is consistent with one of the earliest observations by Taylor [Proc. R. Soc. London Ser. A. 124, 243 (1929)] that steady, symmetric laminar flows can be observed over a larger range of Dean number in tightly coiled tubes. However, when an asymmetric perturbation is imposed at the inlet, oscillatory solutions develop even for low Rc, indicating that symmetric two-dimensional solutions are not stable to asymmetric perturbations, as indicated by Winters [K. W. Winters and R. C. G. Brindley (private communication)]. Numerical results are also presented for flow through curved ducts with periodic step changes in curvature.