Broadband noise prediction of fan outlet guide vane using a cascade response function

Abstract

An analytical model of the broadband noise produced by both the interaction of ingested turbulence with a fan rotor blades and the rotor-wake impingement on downstream stator vanes is proposed and detailed. The noise prediction methodology is a strip-theory approach based on a previously published formulation of the three-dimensional unsteady blade loading for a rectilinear cascade. This three-dimensional cascade response applied in each strip combined with an acoustic analogy in an annular duct have been chosen to account for the main three-dimensional effects. To further improve some of the identified limitations of this approach, a correction is added to mitigate the effects of the non-coincidence of the cut-on frequencies of the annular duct modes and of the modes of the rectilinear cascade. A correction of the unsteady blade loading formulation, previously developed in a tonal configuration, is also introduced to account for the dispersion relation of annular duct modes in the rectilinear-cascade model. The model is compared with experimental results of the 22-in source diagnostic test (SDT) fan rig of the NASA Glenn Research Center. A numerical assessment of the simplifications proposed in the model and of the convergence of the truncated sums in spanwise wavenumbers and azimuthal orders of the incident perturbation is carried out. The subcritical gusts are shown to have a crucial effect at low frequencies, whereas they become negligible at higher frequencies. Furthermore, alternative high-frequency formulations lead to a satisfactory accuracy above a Helmholtz number based on the duct radius of 20. The strong reduction in computational time associated with these formulations could justify their use for parametric studies in industrial context. The effect of the turbulence model is also investigated showing the relevance of Liepmann's isotropic model in the SDT case, and a possible strong effect of anisotropy in static tests. Finally, the model is compared with NASA's experimental results for two outlet guide vanes at approach condition, showing a very good agreement upstream, whereas an underestimate of 3–5 dB is observed downstream in the middle frequency range.

Introduction

Turbofan engines with higher bypass ratios ensure improved aircraft performances at lower nominal rotation speed. Both the exhaust velocities of burnt gases and the corresponding jet noise are reduced and the fan-OGV (outlet guide vane) stage becomes a major contributor to the total noise. Modern very-high bypass architectures involve lower fan tip speed, reduced number of blades, selected blade and vane counts, and acoustic liners. This ensures tonal noise reduction and shifts the tone frequencies to lower values associated with weaker loudness. As a result, the broadband noise contribution is expected to become relatively more significant and dedicated prediction schemes are a crucial step to be included in the design cycles, as early and as accurately as possible. More specifically, the present study is dedicated to the prediction of the broadband noise resulting from the impingement of incident turbulence on a blade row either rotating or stationary. Numerical simulations of the turbulent compressible three-dimensional flow around the blades or the vanes could in principle reproduce all sound generation and propagation phenomena accurately, but are still a daunting task for an actual fan (blade span of the order of 1 m and Reynolds number based on the chord length and mean velocity around 10⁶) and are far from being compatible with industrial time constraints. Besides, fan broadband noise prediction requires the whole power spectral density (PSD) of the acoustic power for frequencies ranging up to 10 kHz. Fast-running analytical models then appear to be still more appropriate in an industrial context.

Many studies have been performed over the past 40 years to predict the fan broadband noise caused by turbulence ingestion and wake interactions. Mugridge and Morfey [1], Mani [2], Hanson [3] and Sevik [4], among others, have dealt with the interaction of an incident turbulence with a rotating blade or a stationary vane. Homicz and George [5] extended these works to rotating blades at low frequencies and George and Kim [6] and Amiet [7] at high frequencies. Detailed reviews of the methods and experiments carried out during the seventies on rotor broadband noise were proposed by Cumpsty [8], Brooks and Schlinker [9] and George and Chou [10]. Most of these works were dedicated to open rotors, in particular the main rotor of a helicopter, for which the moderate blade number, often smaller than 10, allows using an isolated-airfoil response function.

Dealing with turbofan engines requires including the in-duct propagation in the prediction methods. Glegg [11], De Gouville [12] and Joseph and Parry [13] developed broadband noise models for ducted fans using Green's function tailored to the duct, and the unsteady blade loadings as acoustic sources. Glegg [11] dealt with the interaction of the rotor blades with the boundary layer of the casing and with the interaction of rotor wakes with the OGV. Yet the blades were assumed to be acoustically compact along the chord. De Gouville [12] resorted to Graham's similarity rules [14] to account for the compressibility effects and the non-compactness of the airfoil to determine the turbulence ingestion noise, while Joseph and Parry [13] used the two-dimensional compressible Amiet's function to predict the noise due to the interaction of the casing boundary layer with the rotor blades. These investigations were based on an isolated-airfoil response function. Nevertheless, current turbofan engines involve higher and higher bypass ratio and chord lengths, and the blade number of the rotor and the vane number of the stator can easily exceed 20 and 50 respectively (e.g. [15]). As a result, cascade effects must also be included in the prediction methods. Ventres et al. [16] were the first to propose a fan broadband noise model for inlet and wake turbulence considering both the duct and the cascade effects. These two aspects are an important theoretical improvement. An in-duct formulation of the acoustic analogy was applied, using the unsteady blade loading as input data, but resorting to a two-dimensional cascade response function. The radial variation of the turbulence was then taken into account by means of a strip theory. More recently Nallasamy and Envia [17] enhanced and coupled this model to a Reynolds-Averaged Navier–Stokes computation to get the turbulence input data for the acoustic model, showing a good agreement with measurements.

However, experimental results (e.g. [18]) show clear evidence of the strong spanwise variation of the turbulence in the rotor wakes developing upstream of the OGV, which may hinder the above two-dimensional decomposition of the impinging turbulence on stator vanes. Another approach then consists in including the spanwise variations of the turbulence by means of a three-dimensional rectilinear-cascade model: incident gusts are three-dimensional but the cascade is still considered rectilinear. Hanson [19] and Hanson and Horan [20] proposed a model for the interaction of a homogeneous or radially inhomogeneous incident turbulence with a rectilinear flat-plate cascade resorting to Glegg's cascade model [21]. Hanson then extended it to swept and leaned cascades [22] and to the broadband noise of a complete fan stage [23]. Glegg [24] and Glegg and Walker [25] accounted for the duct wall in the unwrapped configuration giving the exact solution to this approximate problem by means of a cosine functions basis. Evers and Peake [26] finally extended the method of Hanson and Horan [20] to include blade geometry effects. These predictions agreed quite well with the available experimental results despite the crucial assumption of rectilinear cascade [20]. Cheong et al. [27] again using a rectilinear-cascade model based on Smith's two-dimensional theory [28], pointed out a critical frequency, below which cascade effects are important, and only a part of the turbulent wavenumbers contribute to the resulting noise. Above that frequency, cascade effects can be neglected and the whole incident turbulence contributes. Jurdic et al. [29] predicted the rotor/stator interaction noise with this model using RANS data as inputs. Hanson's model and similar ones can be declined in a strip theory to take the radial variation of the geometry of an actual annular cascade into account. Yet, it directly applies to the radiated field and associated acoustic power, and does not rely on the unsteady blade loading as in the previous methods. As a result, the propagation in an annular duct cannot be accounted for.

Finally a significant improvement in fan broadband noise predictions has been recently achieved through three-dimensional unsteady linearized-Euler (LEE) simulations, fully accounting for the actual blade geometry. For instance, Atassi and Vinogradov [30], [31] and Atassi and Logue [32] proposed a very accurate fan broadband interaction noise method based on a previously developed model [33]. This approach noticeably accounts for the three-dimensional effects of the actual geometry, swirling mean flow and three-dimensional turbulence excitation and points out the importance of these parameters [33].

The present paper describes an analytical model of the broadband noise produced by turbulence in a rotor–stator arrangement. Both the interaction of ingested turbulence with the rotor blades and the rotor-wake impingement on downstream stator vanes are addressed. The model is a strip-theory application [34] based of a previously published formulation of the unsteady blade loading for a rectilinear cascade [35] and of the subsequent developments by the authors [34]. More precisely, Glegg's analytical formulation [21] has first been extended to provide closed-form expressions of the acoustic field valid inside the inter-blade channels, and of the unsteady-blade loading [35]. A wavenumber correction has also been proposed to include some of the three-dimensionality inherent to the annular configuration into the unwrapped description of a cascade strip in Cartesian coordinates [34]. This makes the unsteady blade loading calculated on each strip an equivalent dipole source distribution in an acoustic analogy formulation inside an annular duct. Preliminary broadband noise prediction issues have been addressed by Posson and Roger [36] and Posson et al. [37] to assess the effect of simplifying assumptions and the role of specific corrections to account for the main three-dimensional effects. The model was compared with the reference three-dimensional LEE computation of Atassi et al. [33] and with Logue and Atassi's updated version of the linear cascade model of Atassi and Hamad [38]. Posson and Roger [39] also carried out a dedicated experiment involving a turbulence grid upstream of a stationary cascade mounted at the exit section of an open-jet anechoic wind-tunnel. The model was found in a rather good agreement with the experiment in an extended low and middle frequency range. Further validations in configurations closer to a real engine were however recognized as necessary. The final version of the broadband noise model is described in detail in Section 2. The predictions are then compared, in Section 3, with the experimental data of the 22-in source diagnostic test (SDT) fan rig of the NASA Glenn Research Center [40], [41], [42], [43], [44], [45], [46]. Both numerical assessment and experimental validation are achieved.