Frequency-domain prediction of broadband inflow noise radiating from a finite-thickness airfoil

https://doi.org/10.1016/j.jweia.2021.104618Get rights and content

Highlights

  • A semi-analytic model is developed for the prediction of broadband inflow noise of an airfoil. .

  • The model can account for airfoil geometry effects in generation and propagation phases of airfoil inflow noise. .

  • An acoustic transfer function is derived to consider airfoil geometry effects in the propagation phase.

  • The algebraic attenuation model is employed to consider airfoil geometry effects in the generation phase.

  • Predictions using the semi-analytic model provide close agreements with the measurements.

  • The present model reproduces a more realistic directivity of broadband inflow noise. .

Abstract

The purpose of this study is to develop a semi-analytic model for the prediction of broadband inflow noise of an airfoil in the frequency-domain, which can account for the effects of real airfoil geometry on the radiated sound pressure field in both phases of propagation and generation of aerodynamic sound. First, an acoustic transfer function is analytically derived to account for airfoil geometry’s effects in the propagation phase. The algebraic attenuation model is then combined to consider the effects of incident turbulence distortion in the generation phase. Predictions using the semi-analytic model provide much closer agreements with the measurements than the analytic model in a broader range of frequency and mean flow speed. Besides, the model is shown to reproduce a more realistic directivity of broadband inflow noise by including a drag-type loading source due to the finite thickness of an airfoil as well as a lift-type one.

Introduction

Broadband inflow noise is generated due to the scattering of the incident turbulence by a foreign body. The inflow broadband noise of a rotor is generated by the interaction of incident atmospheric turbulence (or wake turbulence from a foreign body upstream) with rotating rotor blades. Low-frequency noise radiating from a wind turbine is mainly due to the inflow noise, and its long-distance propagation induces the environmental noise issue of wind turbines (Jung et al., 2008; Lee and Cheong, 2014; Cheong and Joseph, 2014; Laratro et al., 2014).

Since it is still challenging due to enormous numerical cost to tackle these problems with the high-resolution computational aeroacoustic techniques such as the DNS and LES methods, analytic models (Lee and Cheong, 2014; Cheong and Joseph, 2014; Wasala et al., 2015) have been developed as an alternative. The extension of the two-dimensional analytic models to the three-dimensional rotor is best suited to large wind turbines because their slow rotation speed and large span satisfy the condition necessary for the extension: the angular frequency of rotation is very small compared to the angular acoustic frequency, i.e., ω/Ω⪢1, which allows the effects of rotation as a series of translations over an infinitesimal distance (Lowson, 1965).

The development of more reliable analytic models needs the physics-based modeling of the following three-sequential mechanism of inflow broadband noise: statistical characteristics of an incident turbulence field onto a foreign body, aerodynamic responses of the foreign body by its interaction with the incident field, and acoustic wave propagation to an observing point from these loading sources.

The incident turbulence field is frequently regarded as a homogenous and isotropic field for simplification of the source model for broadband inflow noise. Most wind tunnel experiments have utilized a turbulent grid within a nozzle to generate the turbulent field, and turbulent intensity and integral length scale are then determined to characterize the incident turbulence field (Paterson and Amiet, 1976). The von Karman and Liepmann turbulent energy spectrum are frequently used to determine the statistical characteristics of the turbulent field by using its intensity and length scale. Since the inflow airfoil noise can be understood as the resultant sound field induced by the interaction of local homogeneous and isotropic incident turbulences with a leading edge of an airfoil, various studies focused on the aerodynamic response and aeroacoustic characteristics of airfoils (Amiet, 1975a; Gershfeld, 2004; Lysak et al., 2013; Kim et al., 2015a, Kim et al., 2015b).

Amiet (1975a) developed an inflow broadband noise model based on the thin airfoil theory (1975b), where Curle’s acoustic analogy (Curle, 1955) was applied to derive the analytic solution for acoustic pressure radiating from aerodynamic loading of a flat plate. In this approach, an analytic integration is made with a large span approximation, which simplifies the derivation of a closed-form solution for acoustic pressure. The loading by the scattering of the turbulent fields near the airfoil leading edge is computed by using Schwarzschild’s solution for the convective wave equation with the half-plane condition (Amiet, 1975b). Comparing the measured data with the predictions obtained from the thin airfoil model, Paterson and Amiet showed that the thin airfoil model overestimates inflow noise in the high-frequency range at low Mach number (Paterson and Amiet, 1976).

The thin airfoil model cannot consider the effects of airfoil geometry on the radiated broadband noise. The effects of airfoil geometry can be understood in the two sequential phases: noise generation and propagation phases. The former is related to the aerodynamic responses of an airfoil subject to the incident turbulence field, and the latter is related to acoustic wave propagation to an observing point from a loading source. The goal of the present study is to improve the thin airfoil model by considering the effects of airfoil geometry in these two phases.

Zhou (2004) derived a frequency-domain formulation for predicting the broadband self-noise due to an open rotor. The integral formulation was evaluated on the real blade surface rather than on the projected flat plate, thereby considering the effects of airfoil geometry in the propagation phase. An approach similar to that of Zhou (Zhou and Joseph, 2007) is adopted in this study, not for predicting broadband self-noise, but for predicting broadband inflow noise of a real airfoil.

Küçükosman et al. (2018) combined the Amiet’s semi-analytic model with Reynolds-Averaged Navier-Stokes (RANS) computations for the prediction of the trailing-edge noise by a NACA0012 airfoil and showed that the combined models provided the predicted results in closer agreement with the measured ones than the models based on flat-plate boundary layer data. In a similar way, the accuracy of inflow broadband noise models based on the thin airfoil model needs to be improved by considering the aerodynamic response of a real airfoil in the generation phase. Distortion of the incident turbulent field near the airfoil leading edge with finite curvature was found to cause the reduction of noise in a high-frequency range, and several studies were conducted to correct the overestimation (Gershfeld, 2004; Lysak et al., 2013; Kim et al., 2015a, Kim et al., 2015b). Gershfeld (2004) suggested a correction formula in the form of an exponential function to account for the reduction of inflow noise in the high-frequency range due to airfoil geometry and showed that the predictions made using the corrected formula more closely follow the experimental data by Paterson and Amiet (Paterson and Amiet, 1976). The main argument of the correction formula is the Strouhal number computed using the airfoil thickness and the convective velocity of eddies. Lysak et al. (2013) computed unsteady lift on the Karman-Treffz airfoils, whose leading edge and thickness are similar to NACA65-012, by employing the gust in the form of the step function. They tried to express the unsteady lift of the Karman-Treffz airfoils with the Sears function multiplied by a correction function, which was obtained using the curve fitting of computed lifts that are primary sources of the inflow noise. The correction function is an exponential function of the Strouhal number. Kim et al., 2015a, Kim et al., 2015b utilized the high-order computational aeroacoustic (CAA) techniques to estimate the effects of the thickness and leading-edge sharpness of an airfoil on the acoustic power level due to airfoil inflow broadband noise. Using the numerical simulation results for airfoils with different thicknesses, they suggested a correction function to account for the effects of airfoil thickness on acoustic fields. The correction function was expressed with airfoil thickness, Mach number, and leading-edge radius, thereby being used to correct the sound power levels obtained from the inflow noise model developed by Blandeau et al. (2010).

In this study, the inflow broadband noise model in a frequency-domain (real airfoil model) is developed to account for the effects of real airfoil geometry in both generation and propagation phases of inflow noise. The original contribution of the present study is twofold. First, an analytic acoustic transfer function is derived for predicting acoustic pressure at the observed point in the acoustic field due to the aerodynamic loading on a finite element of an airfoil surface to account for the geometry effect in the propagation phase. The drag-dipolar directivity of the acoustic pressure is found to be clearly reproduced by using the analytic real airfoil model. Second, the algebraic correction formula by Kim et al., 2015a, Kim et al., 2015b is combined with the analytic real airfoil model to consider the effects of real airfoil geometry in the generation phase. Acoustic predictions made using the semi-analytic real airfoil model show much closer agreements with the experiments in wider ranges of frequency and mean flow Mach number.

The analytic solution of broadband inflow noise based on the Green’s function solution is briefly described for calculating acoustic wave propagation due to the aerodynamic loading in Sections 2. In Section 3, the analytic solution of broadband inflow noise described in Section 2 is extended for considering real airfoil geometry by introducing a numerical acoustic transfer function on a finite element on the airfoil surface. In Section 4, an attenuation function due to the distortion of the incident turbulence field near the leading edge of an airfoil is described and combined with the analytic real airfoil model to consider the effect of real airfoil geometry in the generation phase. In Section 5, the developed analytic and semi-analytic real airfoil models are applied to the benchmark problem where the measured sound pressure spectra are available at various mean flow speeds.

Section snippets

Broadband inflow noise model

For acoustic source term Q(y,τ), the convective wave equation is written as:1a02D2Dτ2p2p=Q(y,τ)where D2/2=(∂/∂τ+U∂/∂y1)2.

pis acoustic pressure, τ is retarded time, a0 means the speed of sound, U indicates mean flow speed in the y1 direction and Q is an acoustic source term. Equation (1) can be re-written by using a linear differential operator L as:L(y,τ)p(y,τ)=Q(y,τ),whereL(y,τ)=1a02D2Dτ22.

The integral form of acoustic-field solution to Eq. (2) can be obtained by using the Green’s

Acoustic transfer function

An analytic solution for the acoustic transfer function of Eq. (20) can be computed by summation form as shown in:HA(x,y,kS,ω)=n=1N{FnG(x,yn,ω)FnL(yn,kS,ω)},whereFnG(x,yn,ω)=Nn(yn)G¯(x,y,ω)yi,G¯(x,y,ω)yi={ejμRM4πR(ynx1R2+jμ(ynx1R+M)),i=1ejμRM4πR(β2Rjk)(ynxiR),i=2or3FnL(yn,kS,ω)=Snht(y1,kS,ω)ejkSy2dSn(y),ht(y1,kS,ω)=ejkμy1y1,

andkμ={ωa0β2(1kk2ks2β2M),ωa02β2>kS2jωa0β2(1kks2β2k2+jM),ωa02β2<kS2.

In Eqs. (26)–(31), k is the acoustic wavenumber, Nn denotes the unit outward vector

Attenuation model due to incident turbulence distortion

In this section, the real airfoil model described in Section 3 is combined with an attenuation function to consider the effects of the distortion of an incident turbulence field in the vicinity of the airfoil leading edge in the sound generation phase.

Kim et al. (2015) performed an extensive parametric study on inflow noise from symmetric airfoils interacting with inflow turbulence using an extensive parametric study using a high-order computational aeroacoustic technique combined with a

Correction of wind tunnel shear layer

Before the developed model is applied to the benchmark problem (Paterson and Amiet, 1976) for validation, the effects of the wind tunnel shear layer on acoustic propagation need to be accounted for and included in the model. Veltin et al. (2010) explained the effects of the shear layer, which is summarized and utilized in this section.

Fig. 2 schematically describes acoustic wave propagation from a source to a microphone in a wind tunnel. Through the shear layer, acoustic waves satisfy the

Conclusions

In this study, a semi-analytic model in the frequency-domain was developed to predict inflow broadband noise caused by the interaction of incident turbulence with a real airfoil of finite-thickness. First, the analytic transfer function was derived for acoustic pressure at a receiver point due to aerodynamic loadings on a single finite element of airfoil surface to consider the effects of the airfoil geometry on the propagation of acoustic pressure wave. It was then extended into the

Author contributions

C.C. provided the basic idea for this study and worked on the analysis of the simulation results. G.-S. L. derived the analytic formulations and carried out the numerical simulations, and worked on the analysis of numerical results.

Funding

This research was funded by the Ministry of Trade, Industry & Energy, Republic of Korea (No. 20183010025530).

CRediT authorship contribution statement

Gwang-Se Lee: Formal analysis, Methodology. Cheolung Cheong: Formal analysis, Writing – original draft.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work was supported by Korea Institute of Energy Technology Evaluation and Planning(KETEP) grant funded by the Korea government(MOTIE) (20183010025530, Development of Smart O&M Platform for Wind Turbine)

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