On a uniformly valid analytical rectilinear cascade response function

Abstract

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.

Experimental Validation of a Cascade Response Function for Fan Broadband Noise Predictions

“…However, using Richardson's procedure together with 100 terms in the matrix gives well-converged results. The good accuracy well below the low-frequency limit of a two-step Schwarzschild procedure has already been outlined in the limit case of a reduced cascade effect [19].…”

“…The analytical model for the broadband noise produced by the impingement of turbulence on a cascade is based on a previous formulation for an unwrapped cascade [18,19], following Glegg's rectilinear-cascade response [20]. The model determines closedform expressions for the induced unsteady loads.…”

Evaluation of a Cascade-Based Acoustic Model for Fan Tonal Noise Prediction

“…For instance, the model from Ventres et al [7], updated by Meyer and Envia [8] and by Nallasamy and Envia [9], deals with a two-dimensional (2-D) vortical excitation (no radial wave number) impinging on a vane cascade. Recently, Posson et al [10,11] proposed a model for rotor-stator interaction considering skewed (3-D) gusts and a 3-D analytical cascade response. Whereas this model has only been used for fan broadband noise applications [12], the current study focuses on the evaluation of this model for fan tonal noise prediction.…”

“…Nevertheless, technological effects have a significant influence on fan tonal noise prediction, as shown for instance by Peake and Kerschen [13] concerning the mean vane loading and by Evers and Peake [14] for the vane camber. The present study aims at evaluating the effects of some assumptions of the model of Posson et al [10] both on the acoustic sources computation and on the fan-noise prediction. For this purpose, two configurations will be investigated: a rather academic configuration of an annular flat-plate cascade that is close to the one of the model, and a more realistic case of a lowpressure axial compressor stage.…”

Effect of Distortion on Turbofan Tonal Noise at Cutback with Hybrid Methods

“…Hybrid methods stem from the acoustic analogy introduced by Lighthill [1] where source generation and noise propagation are separated steps. Acoustic sources can be extracted from a computational fluid dynamics (CFD) simulation or can be evaluated with analytical models, initially developed for isolated airfoils [2,3] and now extended to account for geometrical and cascade effects [4][5][6][7][8][9][10][11]. These sources can then be propagated either numerically by a computational aeroacoustics (CAA) simulation or analytically by an acoustic analogy.…”