Nonlinear evolution of vortical disturbances entrained in the entrance region of confined flows

The nonlinear evolution of free-stream vortical disturbances entrained in the entrance region of confined flows---pipe and channel flows---is investigated using asymptotic and numerical methods. The focus is on low-frequency disturbances that induce streamwise-elongated structures and on Reynolds numbers for which the flow is linearly stable but undergoes intense algebraic growth of disturbances. The disturbance flow along the entrance is generated by free-stream disturbances at the pipe and channel inlets, with their amplitudes being sufficiently intense for nonlinear interactions to occur. The formation and evolution of the disturbance flow are described by the nonlinear unsteady boundary-region equations, derived and solved numerically herein in the pipe and channel flow geometries for the first time. Matched asymptotic expansions are employed to construct appropriate initial conditions, and the initial-boundary value problem is solved numerically by a marching procedure. A parametric study is conducted to examine the effects of the Reynolds number and the characteristics of the inlet disturbances on the nonlinear flow evolution. Numerical results show the stabilising effect of nonlinearity on the intense algebraic growth of the disturbances and an increase of the wall-shear stress due to the nonlinear interactions in both flows. Streamwise elongated pipe- and channel-entrance nonlinear structures (EPENS and ECENS) occupying the whole cross-section are discovered. EPENS possessing a rotational symmetry comprise high-speed streaks near the wall, and low-speed streaks centred around the pipe core. These distinct structures display a striking resemblance to nonlinear travelling waves found numerically and observed experimentally in fully developed pipe flow. ECENS are characterised by low- and high-speed streaks and streamwise vortices whose cores are centred at the channel centreplane. A rotational symmetry with respect to the vortex centres is identified for ECENS. For sufficiently large inlet amplitudes, their dynamics becomes largely independent from the symmetrical properties of the inlet disturbances.