Nonlinear evolution of vortical disturbances entrained in the entrance region of a channel

The nonlinear evolution of free-stream vortical disturbances entrained in the entrance region of a channel is investigated using asymptotic and numerical methods, building on the linear framework developed by Ricco & Alvarenga (2021 J. Fluid Mech. , vol. 927, A18). The focus is on low-frequency disturbances that induce streamwise-elongated structures at Reynolds numbers for which the entrance flow is locally stable according to classical linear stability theory. The perturbation flow along the channel entrance is generated by free-stream vortical disturbances located at the channel inlet. These disturbances are symmetric or antisymmetric with respect to the centreplane and their amplitude is sufficiently intense to provoke nonlinear interactions within the channel. The formation and evolution of the perturbation flow are described by the nonlinear unsteady boundary-region equations. Combined with physically realistic initial conditions, the resulting initial-boundary-value problems are solved numerically using a streamwise integration method. A parametric study is conducted to elucidate how the nonlinear channel flow is influenced by the Reynolds number and the inlet-disturbance properties, i.e. the amplitude and the streamwise, wall-normal and spanwise wavelengths. Nonlinearity is found to stabilise the intense algebraic growth and to drive the formation of elongated channel-entrance structures that span the entire cross-section. These structures, characterised by low- and high-speed regions and streamwise vortices, meander along the streamwise direction and persist even when the base flow is fully developed. They exhibit a half-turn rotational symmetry with respect to the vortex centres. These properties emerge downstream regardless of the symmetry of the initial perturbation flow, provided nonlinear interactions are sufficiently intense. The occurrence of travelling waves is detected sufficiently downstream, and their similarity to those found in the fully developed region by other researchers is discussed. Our results show good agreement with theoretical predictions, numerical results and experimental measurements for both the mean flow and the perturbation flow.