Adjoint design methods for nozzles for reduced noise in high-speed jets

This paper describes the use adjoint methods for nozzle design and control optimization to reduce noise in supersonic jets operating at off-design conditions. The motivation for the work is the development of an optimization procedure for a new jet noise reduction concept that involves flow injection into the diverging section of a convergent-divergent nozzle. The noise reduction method and some preliminary flow and acoustic measurements are described. Noise reductions of more than 5dB in mixing noise as well as reductions in broadband shock associated noise are demonstrated. Two adjoint optimization methods are then described. The first example, provided to fix the ideas of adjoint design, involves changing the nozzle contours to achieve a desired pressure distribution in the nozzle. The second example involves the determination of the optimum distribution of blowing and/or suction in the diverging section of the nozzle to achieve a minimization of the shock strength. Following a description of the noise reduction concept and some preliminary results, a general review of adjoint methods in design is given. In adjoint design, a cost function is minimized within defined constraints. These include the requirement that the flow satisfies the flow conservation equations and boundary conditions as well as other smoothness constraints. The sensitivity of the cost function to the design parameters can be obtained from a solution of an adjoint problem. The adjoint problem formulation for two example approaches is given. Results are given in two cases for convergent-divergent nozzles with jets operating at different pressure ratios including an over-expanded condition, which is typical of high performance military aircraft engines on take-off.