Sensitivity of aerodynamic forces in laminar and turbulent flow past a square cylinder

We use adjoint-based gradients to analyze the sensitivity of the drag force on a square cylinder. At Re = 40, the flow settles down to a steady state. The quantity of interest in the adjoint formulation is the steady asymptotic value of drag reached after the initial transient, whose sensitivity is computed solving a steady adjoint problem from knowledge of the stable base solution. At Re = 100, the flow develops to the time-periodic, vortex-shedding state. The quantity of interest is rather the time-averaged mean drag, whose sensitivity is computed integrating backwards in time an unsteady adjoint problem from knowledge of the entire history of the vortex-shedding solution. Such theoretical frameworks allow us to identify the sensitive regions without computing the actually controlled states, and provide a relevant and systematic guideline on where in the flow to insert a secondary control cylinder in the attempt to reduce drag, as established from comparisons with dedicated numerical simulations of the two-cylinder system. For the unsteady case at Re = 100, we also compute an approximation to the mean drag sensitivity solving a steady adjoint problem from knowledge of only the mean flow solution, and show the approach to carry valuable information in view of guiding relevant control strategy, besides reducing tremendously the related numerical effort. An extension of this simplified framework to turbulent flow regime is examined revisiting the widely benchmarked flow at Reynolds number Re = 22 000, the theoretical predictions obtained in the frame of unsteady Reynolds-averaged Navier–Stokes modeling being consistent with experimental data from the literature. Application of the various sensitivity frameworks to alternative control objectives such as increasing the lift and reducing the fluctuating drag and lift is also discussed and illustrated with a few selected examples.