Flow in a weakly curved square duct: Assessment and extension of Dean's model

Abstract

The simplified model by W. R. Dean, based on a low-curvature assumption, provided an early understanding of the laminar flow in curved ducts. However, most of the following studies relied on computer simulations of the complete curvilinear Navier-Stokes equations controlled by two nondimensional parameters, the curvature ratio and the Reynolds number. In the present article an extended version of Dean's model is used and compared to existing results. The set of equations is unsteady, parabolic in a streamwise direction, expressed in Cartesian coordinates, and contains a single control parameter, namely, the Dean number. The equations are identical to those used for the Görtler instability in boundary layer flows. Nonetheless, their extension to duct flows remains to be validated, which is the main purpose of the present article. The model satisfactorily reproduces benchmark results in the literature. In particular we retrieve the successive bifurcations between steady, unsteady, and chaotic regimes for 2D flows. The model also reproduces the development of three-dimensional flow in an elbow with a curvature radius equal to 15.1 times the square duct width. In addition, the present results confirm the Dean number as the single control parameter for laminar flows in a weakly curved ducts.