Stabilising pipe flow by a baffle designed using energy stability

Previous experimental (Kühnen et al., Flow Turb. Combust., vol. 100, 2018, pp. 919–943) and numerical (Marensi et al., J. Fluid Mech., vol. 863, 2019, pp. 850–875) studies have demonstrated that a streamwise-localised baffle can fully relaminarise pipe flow turbulence at Reynolds numbers of O(104) . Optimising the design of the baffle involves tackling a complicated variational problem built around time stepping turbulent solutions of the Navier–Stokes equations which is difficult to solve. Here instead, we investigate a much simpler ‘spectral’ approach based upon maximising the energy stability of the baffle-modified laminar flow. The ensuing optimal problem has much in common with the variational procedure to derive an upper bound on the energy dissipation rate in turbulent flows (e.g. Plasting & Kerswell, J. Fluid Mech., vol. 477, 2003, pp. 363–379) so well-honed techniques developed there can be used to solve the problem here. The baffle is modelled by a linear drag force −F(x)u (with F(x)≥0 ∀x ) where the extent of the baffle is constrained by an Lα norm with various choices explored in the range 1≤α≤2 . An asymptotic analysis demonstrates that the optimal baffle is always axisymmetric and streamwise independent, retaining just radial dependence. The optimal baffle which emerges in all cases has a similar structure to that found to work in experiments: the baffle retards the flow in the pipe centre causing the flow to become faster near the wall thereby reducing the turbulent shear there. Numerical simulations demonstrate that the designed baffle can relaminarise turbulence efficiently at moderate Reynolds numbers ( Re≤3500 ), and an energy saving regime has been identified. Direct numerical simulation at Re=2400 also demonstrates that the drag reduction can be realised by truncating the energy-stability-designed baffle to finite length.