A linearized approach for the control of aerodynamic forces in flow past a square cylinder

The analysis of Strykowski & Sreenivasan (JFM, 1990) provides experimental evidence that a small control cylinder suitably positioned in the wake of a main cylinder can alter vortex shedding close to the instability threshold. This problem was addressed from a theoretical perspective by Hill, who used adjoint-based gradient to compute the sensitivity of the global instability mode correlated to the shedding activity, and thereby retrieved the experimental sensitivity regions without knowledge of the actual controlled states. Such an approach is an attractive alternative to bottleneck “trial and error” procedures in that it allows spanning quickly all possible positions of the control cylinder without ever calculating any controlled state. It has sparked interest as a means to gain beforehand valuable information regarding the most sensitive regions for open-loop control based on the underlying physics. The present research aims at predicting similarly the optimal placement of the control cylinder in the attempt to modify the aerodynamic forces. The main focus is on drag of a square cylinder, intended to serve as a testbed for developing the related methodology. We compute the drag variation caused by a small control cylinder whose diameter is 1/10 that of the main cylinder, whose presence in the flow is modeled by a pointwise reacting force. Calculations are performed for two values of the Reynolds number. The first one Re = 40 is subcritical in the sense of bifurcation theory, as the flow settles down to a steady state. All results are presented in terms of maps of the steady asymptotic drag (i.e., the value reached after the initial transient), whose sensitivity is computed solving a steady adjoint problem from knowledge of the base solution. The second one Re = 100 is supercritical as the flow conversely develops to the time-periodic, vortex-shedding state. Results are rather presented in terms of maps of the time-averaged mean drag, whose sensitivity is computed integrating backwards in time an unsteady adjoint problem from knowledge of the DNS solution.