The Use of an Adjoint Method for Optimization of Blowing in a Convergent-Divergent Nozzle

Abstract

This paper describes an adjoint optimization method. It is used to determine the blowing distribution in the diverging section of a convergent-divergent nozzle that gives a desired pressure distribution in the nozzle. The study is motivated by a noise reduction concept described by Morris et al. [11, 12] in which distributed blowing in the nozzle divergent section is used to repro- duce the effect of the corrugated seals developed by Seiner et al. [15, 14]. Both approaches reduce broadband shock-associated noise by allowing the nozzle to operate closer to an on-design condition. In addition, streamwise vortices are produced that break up the large scale structures in the jet and reduce mixing noise in the peak noise radiation direction. The design of the distributed blowing involves many parameters: Both in the injector geometry and operating conditions. This design problem is ideally suited to the adjoint optimization method. The present paper provides a framework for this design process. The paper develops both the direct and adjoint problems and their associated boundary conditions. Two simple example problems are described. In the first, a prescribed blowing distribution is imposed and the resulting pressure distribution in the nozzle is calculated. This pressure distribution is used as the target or desired distribution and the required distributed blowing to achieve this target, starting from zero blowing, is determined using the adjoint method. In the second case blowing is introduced in a convergent-divergent nozzle containing a shock. The adjoint method is used to determine the blowing distribution required to eliminate the shock.