Thermoacoustic Shape Optimization of a Subsonic Nozzle

Indirect combustion noise originates from the acceleration of non-uniform temperature or high vorticity regions when convected through a nozzle or a turbine. In an recent contribution (Giauque et al., JEGTP, 2012), the authors have presented an analytical thermoacoustic model providing the indirect combustion noise generated by a subcritical nozzle when forced with entropy waves. This model explicitly takes into account the effect of the local changes in the cross-section area along the configuration of interest. In this article, the authors introduce this model into an optimization procedure in order to minimize or maximize the thermoacoustic noise emitted by arbitrary shaped nozzles operating under subsonic conditions. Each component of the complete algorithm is described in details. The evolution of the cross-section changes are introduced using Beziers splines which provide the necessary freedom to actually achieve arbitrary shapes. Beziers poles coordinates constitute the parameters defining the geometry of a given individual nozzle. Starting from a population of nozzles of random shapes, it is shown that a specifically designed genetic optimization algorithm coupled with the analytical model converges at will toward a quieter or noisier population. As already described by Bloy (JFM, 1979), results therefore confirm the significant dependence of the indirect combustion noise with respect to the shape of the nozzle, even when the operating regime is kept constant. It appears that the quietest nozzle profile evolves almost linearly along its converging and diverging sections leading to a square evolution of the cross-section area. Providing insight in the underlying physical reason leading to the difference in noise emission between two extreme individuals, the integral value of the source term of the equation describing the behavior of the acoustic pressure of the nozzle is considered. It is shown that its evolution with the frequency can be related to the global acoustic emission. Strong evidence suggest that the noise emission increases as the source term in the converging and diverging parts less compensate each other. The main result of this article is the definition and proposition of an Acoustic Emission Factor which can be used as a surrogate to the complex determination of the exact acoustic levels in the nozzle for the thermoacoustic shape optimization of nozzle flows. This Acoustic Emission Factor, much faster to compute, only involves the knowledge of the evolution of the cross-section area as well as the inlet thermodynamic and velocity characteristics to be computed.