Onset of three-dimensionality, equilibria, and early transition in flow over a backward-facing step

Abstract

A numerical study of three-dimensional equilibria and transition to turbulence in flow over a backward-facing step is performed using direct numerical solution of the incompressible Navier-Stokes equations. The numerical method is a high-order-accurate mixed spectral/spectral-element method with efficient viscous outflow boundary conditions. The appearance of three-dimensionality in nominally two-dimensional geometries is investigated at representative Reynolds numbers ranging from the onset of three-dimensional bifurcation to later transitional stages. Strongly three-dimensional regions are identified through standard correlation coefficients and new three-dimensionality indices, as well as through instantaneous and time-average streamline patterns and vorticity contours. Our results indicate that onset of three-dimensionality occurs at the boundaries between the primary and secondary recirculating zones with the main channel flow, the latter being the most stable flow component. There is. therefore, strong secondary instability in the shear layers, mainly due to the one emanating from the step corner.The flow further downstream is excited through the action of the upstream shear layers acquiring a wavy form closely resembling Tollmien–Schlichting waves both spatially and temporally with a characteristic frequency f1; upstream, at the shear layer another incommensurate frequency, f2, is present. The two-frequency flow locks-in to a single frequency if external excitations are imposed at the inflow at a frequency close to f1 or f2; the smaller amplitude excitations, however, may cause a strong quasi-periodic response. Such excitations may significantly increase or decrease (by more than 20%) the length of the primary separation zone XR at lock-in or quasi-periodic states. The equilibrium states resulting from the secondary instability at supercritical Reynolds numbers produce a flow modulated in the spanwise direction, with corresponding variations in the reattachment location XR. While three-dimensionality explains partially the discrepancy between numerical predictions and experimental results on XR at higher Reynolds number Re, the main source of discrepancy is attributed to the inflow conditions, and in particular to external disturbances superimposed on the mean flow, the latter being the main reason also for the somewhat earlier transition found in laboratory experiments.