## Section: New Results

### Multi-point optimization of a time-periodic system of pulsating jets

Participants : Jean-Antoine Désidéri, Régis Duvigneau.

A multi-point optimization exercise governed by the time-dependent compressible Navier-Stokes equations has been solved based on the sensitivity analysis (see above). A system was considered consisting of three pulsating jets acting on a flat-plate boundary layer. As it is well-known, the flow mixing by the jets has the effect of reducing the drag, as this was confirmed by the simulation of the flow in the somewhat arbitrary initial setting of the jets. Then, positions and pulsation frequencies of the jets have been maintained fixed, while their amplitudes and phases, six parameters in total, have been optimized to minimize the drag force. The finite-volume simulation of the time-periodic flow provides the drag force as a function of time over a large number of timesteps (800 for an accurate description of a period). The sensitivity analysis simultaneously provides the derivatives of drag with respect to the six design parameters. These derivatives were averaged over 20 distinct time-intervals, thus yielding 20 averaged gradient vectors of dimension 6. The MGDA was then used to define a descent direction common to the 20 vectors, a descent step was applied to the design parameters, and the process was continued iteratively.

The experiment confirmed the possibility to reduce the drag force at all times of the period, and not only in the average. In contrast, using the average gradient to define the direction of search resulted in a more important reduction of the average drag but at the cost of an increase of drag in a critical portion of the time period. Hence our optimization algorithm is more versatile and powerful than one aiming at minimizing purely statistical functions obtained by time averaging. We also demonstrated the possibility to optimize over a subinterval of the time interval.