Parametric resonance in unsteady watertable flow

The stability of unsteady open-channel flow down an inclined plane is studied using an iterative approach based on the direct and adjoint stability equations combined with a physically justified energy measure. An efficient parametric resonance mechanism has been identified between the exogenous base-flow oscillations and the intrinsic frequencies of streamwise disturbance vortices. This resonance results in strong amplification over a substantial range of the governing parameters, favouring streamwise elongated structures. The optimal frequency for a maximal disturbance response can be efficiently approximated from simpler steady calculations; two frequency-selection criteria are given for this purpose. The analysis generalizes earlier work on steady watertable flow and provides an effective framework and starting point for further work on pattern formation in harmonically forced open-channel flows.