In this work we investigate limiting values of the lift and drag coefficients of profiles in the Helmholtz–Kirchhoff (infinite cavity) flow. The coefficients are based on the wetted arc length of profile surfaces. The problem is to find global minimum and maximum values of the drag coefficient CD under a given lift coefficient CL. We reduce the problem to a constrained problem of calculus of variations and solve it analytically. In so doing we do not only determine extremals but also strictly prove that these extremals realize global extrema. The proofs are based on non-trivial application of Jensen's inequality. The solution of the problem allows us to construct the domain of possible variations of coefficients CL and CD and define maximum and minimum values of the lift-to-drag ratios CL/CD for a given CL.

Flow around a cavity is characterized by a self-sustained mechanism in which the shear layer impinges on the downstream edge of the cavity resulting in a feedback mechanism. Direct Numerical Simulations of the flow at low Reynolds number has been carried out to get pressure and velocity fluctuations, for the case of un-actuated and multi frequency actuation. A Reduced Order Model for the isentropic compressible equations based on the method of Proper Orthogonal Decomposition has been constructed. The model has been extended to include the effect of control. The Reduced Order dynamical system shows a divergence in time integration. A method of calibration based on the minimization of a linear functional of error, to the sensitivity of the modes, is proposed. The calibrated low order model is used to design a feedback control of cavity flows based on an observer design. For the experimental implementation of the controller, a state estimate based on the observed pressure measurements is obtained through a linear stochastic estimation. Finally the obtained control is introduced into the Direct Numerical Simulation to obtain a decrease in spectra of the cavity acoustic mode.