Experiments and simulations of a travelling wave state of incompressible Newtonian flow in a curved duct of square cross-section are presented. The travelling wave mode develops from the well-documented steady four-cell flow state and is characterized by oscillations of the two Dean vortices near the centre of the outer wall.
The oscillations were induced by a carefully positioned pin at 5° from the inlet of the curved section along the symmetry line of the cross-section. It was shown that the travelling wave state is characteristic for curved duct flow and that the pin made it possible to observe the oscillations within the 270° long curved duct. Travelling waves were observed at flow rates above Dn = 170 (Dn = Re/(R/a)1/2, where Re is the Reynolds number, R is the radius of curvature of the duct and a is the duct dimension. The curvature ratio, R/a, is 15.1).
If no other disturbances are imposed, the oscillations are the result of the selective amplification of random disturbances in the flow, leading to a broad frequency spectrum. The travelling wave was found to lock in to an imposed periodic disturbance at a selected frequency. The flow structure of the locked state was investigated in detail, using flow visualization and a one-component laser Doppler anemometer to measure streamwise or spanwise velocities. Direct numerical simulations using the package CFDS-FLOW3D are in very good agreement with the experiments and confirm the existence of a fully developed, streamwise-periodic travelling wave state.
The inflow region between the two Dean vortices, which transports low-speed fluid away from the outer wall, creates strongly inflectional spanwise profiles of the streamwise velocity. Similarities with twisting vortices in a curved channel and sinuous oscillations of Görtler vortices show that the travelling waves observed here result from a secondary shear instability of these spanwise inflectional profiles.