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The acoustic and entropy transfer functions of quasi-one-dimensional nozzles are studied analytically for both subsonic and choked flows with and without shock waves. The present analytical study extends both the compact nozzle solution obtained by Marble & Candel (1977) and the effective nozzle length proposed by Stow et al. (2002) and by Goh & Morgans (2011a) to non-zero frequencies for both modulus and phase through an asymptotic expansion of the linearized Euler equations. It also extends the piecewiselinear approximation of the velocity profile in the nozzle proposed by Moase et al. (2007) to any arbitrary profile or equivalently any nozzle geometry. The equations are written as a function of three variables, namely the dimensionless mass, total temperature and entropy fluctuations, yielding a first order linear system of differential equations with varying coefficients, which is solved using the Magnus expansion. The solution shows that both the modulus and the phase of the transfer functions of the nozzle have a strong dependence on the frequency. This holds for both choked flows and subsonic converging diverging nozzles. The method is used to compare two different nozzle geometries with the same inlet and outlet Mach numbers showing that, even if the compact solution predicts no differences between the transfer functions of the two nozzles, significant differences are found at non-zero frequencies. A parametric study is finally performed to calculate the indirect to direct noise ratio for a model combustor, showing that this ratio decreases at higher frequencies.

The present paper is a state-of-the-art of a special class of analytical models to predict the broadband noise generated by thin airfoils in a flow, either clean or disturbed. Three generating mechanisms are addressed, namely the noise from the impingement of upstream turbulence called turbulence-interaction noise, the noise due to the scattering of boundary-layer turbulence as sound at the trailing edge for an attached flow called trailing-edge noise, and the noise generated due to the formation of a coherent vortex shedding in the near wake of a thick trailing edge, called vortex-shedding noise. Different analytical models previously proposed for each mechanism are reviewed, as declinations of the same basic approach inherited from the pioneer work performed by Amiet in the seventies and based on an extensive use of Schwarzschild's technique. This choice is only an alternative to other models available in the literature and is made here for the sake of a unified approach. Issues dealing with the input data and related to the practical applications to fan noise predictions are rapidly outlined. The validity of the models is ckeched against dedicated experiments with thin airfoils and the limitations as the real configurations depart from the model assumptions are pointed out.

This paper extends an existing analytical model of the aeroacoustic response of a rectilinear cascade of flat-plate blades to three-dimensional incident vortical gusts, by providing closed-form expressions for the acoustic field inside the inter-blade channels, as well as for the pressure jump over the blades in subsonic flows. The extended formulation is dedicated to future implementation in a fan-broadband-noise-prediction tool. The intended applications include the modern turbofan engines, for which analytical modelling is believed to be a good alternative to more expensive numerical techniques. The initial model taken as a reference is based on the Wiener–Hopf technique. An analytical solution valid over the whole space is first derived by making an extensive use of the residue theorem. The accuracy of the model is shown by comparing with numerical predictions of benchmark configurations available in the literature. This full exact solution could be used as a reference for future assessment of numerical solvers, of linearized Euler equations for instance, in rectilinear or narrow-annulus configurations. In addition, the pressure jump is a key piece of information because it can be used as a source term in an acoustic analogy when the rectilinear-cascade model is applied to three-dimensional blade rows by resorting to a strip-theory approach. When used as such in a true rectilinear-cascade configuration, it reproduces the exact radiated field that can be derived directly. The solution is also compared to a classical single-airfoil formulation to highlight the cascade effect. This effect is found important when the blades of the cascade overlap significantly, but the cascade solution tends to the single-airfoil one as the overlap goes to zero. This suggests that both models can be used as the continuation of each other if needed.

A previous experimental investigation of the broadband self noise radiated by an industrial cambered controlleddiffusion airfoil embedded in an homogeneous flow at low Mach number has been extended to various aerodynamic loadings. The instrumented airfoil is placed at the exit of an open-jet anechoic wind tunnel, with a jet width of about four chord lengths. Sound is measured in the far field at the same time as the statistical properties of the wall-pressure fluctuations close to the trailing edge. A new set of mean wall-pressure data has been collected on this airfoil at a chord Reynolds number of 2.9 × × 10 5 , which provides some insight on the Reynolds-number effect. Two previously investigated flow regimes with different statistical behaviors are investigated by changing the angle of attack from 8 to 15 deg. They respectively correspond to the nearly separated boundary layer with vortex shedding at the trailing edge and to the turbulent boundary layer initiated by a leading-edge separation.