Optimal forcing to destabilise turbulence in a pipe flow

Recent experiments (Kuehnen et al. 2018) have shown that flattening a turbulent streamwise velocity profile in pipe flow destabilises the turbulence so that the flow relaminarises. We show that a similar phenomenon occurs for laminar pipe flow profiles in the sense that the nonlinear stability of the profile increases as it becomes more flattened by a body force. Significant turbulent drag reduction is found even in cases where the amplitude of the forcing is not large enough to avoid transition being triggered. The artificial body force is designed to mimic a baffle used in the experiments of Kuehnen et al. (2018) and the nonlinear stability is measured by the size of the energy of typical disturbances (here taken to be localised random initial conditions) needed to trigger transition. A new fully nonlinear optimisation problem is then constructed, whereby the "minimal forcing", i.e. the forcing characterised by the lowest amplitude or minimum work done, is sought to just destabilise turbulence. The resulting optimal forcing is related to the experiments of Kuehnen et al. (2018) to provide insights and guidelines on how to exploit this promising direction of flow control in the most efficient way.