Disentangling chaotic laminar and turbulent flow components in a weakly curved square duct flow

For straight duct flows, the transition to turbulence results from perturbations of finite amplitude, characteristic of a subcritical bifurcation depending on the Reynolds number, and the resulting dynamics are abruptly chaotic in space and time. On the other hand, for slightly curved duct flows, the laminar solution is subject to successive supercritical bifurcations, from steady state to temporal chaos, for increasing Dean number. The present article investigates the coexistence of these two scenarios. The method is based on the separate computation of the laminar solution and the turbulent deviation. Assuming a relatively weak curvature, the equations are expressed in Cartesian coordinates; the equations for the laminar component are governed by the Dean number, while the turbulent deviation is not directly sensitive to the curvature and governed by the Reynolds number. The separation between the laminar, temporally complex solution and turbulent disturbances is fundamentally different from the Reynolds decomposition and allows us to distinguish the concomitant effects of the curvature and the Reynolds stress. The results are illustrated, for the case of square duct flow, with the secondary flow in accordance with Prandtl's classification and with the head losses.