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The lift on an airfoil at various angles of attack in turbulent flow

Published online by Cambridge University Press:  09 May 2025

Yongfei ZhaoOpen the ORCID record for Yongfei Zhao[Opens in a new window],
Yang YangOpen the ORCID record for Yang Yang[Opens in a new window] and
Mingshui LiOpen the ORCID record for Mingshui Li[Opens in a new window]
Yongfei Zhao
Affiliation:
Research Center for Wind Engineering, Southwest Jiaotong University, Chengdu 610031, PR China
Yang Yang*
Affiliation:
Research Center for Wind Engineering, Southwest Jiaotong University, Chengdu 610031, PR ChinaKey Laboratory for Wind Engineering of Sichuan Province, Chengdu 610031, PR China
Mingshui Li
Affiliation:
Research Center for Wind Engineering, Southwest Jiaotong University, Chengdu 610031, PR ChinaKey Laboratory for Wind Engineering of Sichuan Province, Chengdu 610031, PR China
*
Corresponding author: Yang Yang, yang_yacad@163.com
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Abstract

The lift aerodynamic admittances of an airfoil at different angles of attack (AoAs) in turbulent flow are investigated using a combination of theoretical and experimental approaches. Two theoretical one-wavenumber aerodynamic admittances, namely the Sears and Atassi functions, are reviewed and uniformly normalized for comparison with experimental results. In theory, generalized aerodynamic admittances are generated by introducing the spanwise influence into one-wavenumber aerodynamic admittances. The influence of AoA on generalized aerodynamic admittance includes its effect on both the spanwise influence term and one-wavenumber aerodynamic admittances. The experiment indicates that prior to the prestall region, the increase in the spanwise influence factor correlates with the increase in AoA, with the growth rate of the spanwise influence factor likewise accelerating. The Atassi functions demonstrate that the influence of AoA on one-wavenumber aerodynamic admittances is based on the assumption of full correlation in the spanwise direction. Experimental results suggest that one-wavenumber aerodynamic admittances are inapplicable to actual turbulence; however, the Atassi function accurately represents experimental values at the corresponding AoA when taking into account the spanwise effects.

JFM classification

Aerodynamics: Flow-structure interactions

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JFM Papers
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Journal of Fluid Mechanics , Volume 1010 , 10 May 2025, A45
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Copyright
© The Author(s), 2025. Published by Cambridge University Press

1. Introduction

An airfoil subjected to unsteady flows has consistently been a critical issue in fluid dynamics for both theorists and engineers. The lift aerodynamic admittance, an essential parameter in aerodynamics, characterizes the unsteady lift response of an airfoil encountering gusts. Accurately understanding the lift aerodynamic admittance and the effects of gusts on structures is crucial to determine the gust loading.

According to Jones et al. (Reference Jones, Cetiner and Smith2022), two-dimensional gust encounters include three types: vertical, streamwise and vortex gusts. These classic unsteady aerodynamic theories related to gusts typically involve assumptions such as thin airfoil, small angle of attack (AoA) and small motion via the assumption that the flow remains attached. In this study, AoA refers to the fixed AoA of the airfoil relative to the mean flow direction. For the first type, von Kármán & Sears (Reference von Kármán and Sears1938) initially examined lift fluctuations for a rigid airfoil subjected to a vertical sinusoidal gust and formulated the theoretical expression derived from thin airfoil and potential flow theories, known as the Sears function. The Sears function has been directly experimentally proved by Jancauskas & Melbourne (Reference Jancauskas and Melbourne1986) and Wei et al. (Reference Wei, Kissing, Wester, Wegt, Schiffmann, Jakirlic, Hölling, Peinke and Tropea2019) at low reduced frequencies. It should be noted that the prerequisite for Sears’ problem is that AoA must be equal to zero. This condition makes the streamwise gust unnecessary to be considered for a flat-plate airfoil. However, when AoA is non-zero, the influence of streamwise gust on the lift should be taken into account. Isaacs (Reference Isaacs1945) was the first to model an airfoil at a constant AoA encountering a streamwise fluctuating flow, inspired by rotor blades subjected to a periodic free stream during each blade revolution. The form of wake was then simplified by Greenberg (Reference Greenberg1947). In fact, Greenberg and Isaacs’ results are almost the same for a fixed airfoil in a streamwise sinusoidal flow with small gust ratios. Isaac and Greenberg’s theories both considered the fluctuating part as a function that is only related to time and does not vary along the airfoil. Granlund et al. (Reference Granlund, Monnier, Ol and Williams2014) and Yang et al. (Reference Yang, Li, Ma and Li2017) experimentally investigated the unsteady lift on an airfoil encountering a sinusoidal streamwise gust and found that the small disturbance has a good agreement with the measured values especially at low frequencies.

Goldstein & Atassi (Reference Goldstein and Atassi1976) extended Sears’ theory to include a streamwise velocity component, and Atassi (Reference Atassi1984) formulated a specific lift formula to evaluate the importance of AoA and airfoil camber on gust response. Atassi’s theory is predicated on Sears’s theory, and the assumption of Sears that the flow disturbance must be periodic is unavoidable. Young & Smyth (Reference Young and Smyth2021) confirmed through experiments that within a linear range, the Atassi function is valid for modelling vortex gusts as vertical gusts superimposed with streamwise gust components. Wang et al. (Reference Wang, Feng, Cao and Wang2024) showed that the Atassi function provides effective theoretical guidance at small AoA, but is invalid at larger AoA. The stall condition evidently fails to meet the criteria of Atassi’s issues as in Manar & Jones (Reference Manar and Jones2017). Another restriction of the Atassi function by Jones et al. (Reference Jones, Cetiner and Smith2022) is that the amplitudes of vertical and streamwise disturbances maintain a consistent connection with their reduced frequency. It could be demonstrated to be a physical necessity based on mass conservation, which has been explained by Young & Smyth (Reference Young and Smyth2021). Therefore, the Atassi function is not limited to this kind of gust, but rather this type of gust is the only physically feasible one in a two-dimensional harmonic sinusoidal gust flow with both streamwise and vertical components.

The theoretical and experimental researches above are mainly for the conditions of vertical gusts, streamwise gusts or both. In real-world flow situations, turbulence, as an usual form of vortex gust, is common and unavoidable (Jones et al. Reference Jones, Cetiner and Smith2022). The generalized aerodynamic admittance has been extensively employed in the engineering field under turbulent service conditions, utilizing Sears’ expression together with power spectral density and autocorrelation analysis techniques. According to Liepmann (Reference Liepmann1952, Reference Liepmann1955), the generalized aerodynamic admittance can be defined as the ratio of the lift spectrum to that of the turbulent velocity. However, a large number of experimental studies on airfoils and other streamlined structures have found that the generalized aerodynamic admittance at zero AoA in turbulent flow is dependent upon the flow fields and deviates from the Sears function (Jakobsen Reference Jakobsen1997; Larose & Mann Reference Larose and Mann1998; Scanlan & Jones Reference Scanlan and Jones1999; Li et al. Reference Li, Zhao, Yang and Zhou2022).

To explain the problem above, Graham (Reference Graham1970, Reference Graham1971) derived and numerically calculated the two-wavenumber aerodynamic admittance of a thin plate airfoil at zero AoA in a three-dimensional turbulent gust field. It is a two-wavenumber extension of the Sears function. Filotas (Reference Filotas1969), Mugridge (Reference Mugridge1971) and Blake (Reference Blake1986) subsequently gave the approximate expression, particularly for the factor of the spanwise influence term. Subsequently, Massaro & Graham (Reference Massaro and Graham2015) investigated the effects of the aspect ratio on the generalized aerodynamic admittances of thin plates in turbulence and provided a correction for spanwise effects as a function of the aspect ratio based on this approximation. Inspired by the work of Massaro & Graham (Reference Massaro and Graham2015), Li & Li (Reference Li and Li2017) proposed a fast method to derive the closed-form expression of generalized aerodynamic admittance and Yang et al. (Reference Yang, Li and Liao2019) found that the final lift response could be integrated in the form of the one-wavenumber aerodynamic admittance and the factor of the spanwise influence term. Based on those works about the spanwise effects, Li et al. (Reference Li, Zhao, Yang and Zhou2022) finally found that the flow-dependence of generalized aerodynamic admittance results from the spanwise effects and the Sears function could describe the one-wavenumber aerodynamic admittance of an airfoil at zero AoA well.

However, due to the asymmetry induced by non-zero AoA and the spatial–temporal complexity of turbulence, the influence of AoA on admittances in turbulence predominantly centres on empirical equations and semiempirical analyses (Devinant et al. Reference Devinant, Laverne and Hureau2002; Mannini et al. Reference Mannini, Marra, Pigolotti and Bartoli2017; Li et al. Reference Li, Li, Wu, Li and Yang2023). Extensive research on the effects of high AoA has been conducted under stall conditions, while most of the structures in service under various turbulent conditions such as atmospheric boundary layers do not experience AoA sufficient to induce stall. For non-zero AoA in turbulence, the streamwise velocity fluctuations will definitely contribute to the lift of the airfoil, rather than solely accounting for the vertical components as in Li et al. (Reference Li, Zhao, Yang and Zhou2022). Besides, the two-dimensional theory does not include the effect of turbulent vorticity distortion which occurs in a three-dimensional turbulent flow field interacting with significant gradients of mean flow and is of the same order of magnitude (Goldstein Reference Goldstein1978; Hunt & Carruthers Reference Hunt and Carruthers1990). This effect could appear due to the increasing airfoil AoA. Therefore, further research is required to precisely determine the impact of AoA on the unsteady lift of airfoils under turbulent conditions, thus providing pertinent guidance.

This work aims to assess the impact of AoA on the aerodynamic lift of an airfoil in turbulence by extending a method for calculating the generalized aerodynamic admittance of an airfoil at various AoA in turbulent conditions, without using empirical formulae for generalized aerodynamic admittance. Before comparing theoretical and experimental aerodynamic admittances, the unified normalization of theoretical functions reveals the mechanism of the influence of AoA. Theoretical and experimental results suggest that the one-wavenumber aerodynamic admittance, whether the Sears or Atassi function is used, is not appropriate for practical turbulent flow. Nevertheless, upon evaluating the spanwise effect, generalized aerodynamic admittances utilizing the Atassi function could effectively characterize the impact of AoA on aerodynamic lift in turbulent flows.

2. Theories and methodology

2.1. Theoretical aerodynamic admittances and the unified normalization

Figure 1.

(a) An airfoil at an AoA encountering a sinusoidal vertical gust. (b) An airfoil at an AoA encountering a sinusoidal streamwise gust. (c) Schematic of the problem: an airfoil at an AoA encountering the combined Fourier components of streamwise and vertical fluctuating velocities with spanwise variation.

Section 2.1 provides a concise overview of the theories presented by von Kármán & Sears (Reference von Kármán and Sears1938) and Atassi (Reference Atassi1984), along with unified normalization. In Sears’ model, for a one-dimensional sinusoidal gust field with vertical velocity distribution $w_G=w_0 \mathrm{exp}(\mathrm{i}(\omega _wt-{k}_w^xx))$ , the normalized lift on a flat-plate airfoil is

(2.1) \begin{equation} \frac {L}{\pi\rho c U_0 w_0 \mathrm{exp}(\mathrm{i}\omega _wt)}=S\big(\tilde {k}_w^x\big), \end{equation}

where $U_0$ is the mean flow velocity, $\rho$ denotes the fluid density, $k_w^x=\omega _w/U_0$ indicates the streamwise wavenumber of $w_G$ , $\tilde {k}_w^x=k_w^xc/2$ refers to the reduced wavenumber and $S(\tilde {k}_w^x)$ represents the well-known Sears function.

Based on Sears’ theory, Atassi examines additional streamwise velocity perturbations for a thin airfoil with a small camber $m$ and a small AoA $\alpha _0$ . Considering the vertical velocity distribution $w_G=w_0 \mathrm{exp}[\mathrm{i}(-\omega _wt+{k}_w^xx)]$ , as the fluctuation shown in figure 1(a), and the streamwise velocity distribution $u_G=u_0 \mathrm{exp}[\mathrm{i}(-\omega _ut+{k}_u^xz)]$ , as the fluctuating part show in figure 1(b), the unsteady lift induced by the gust can be formulated through linear superposition, reflecting the Sears lift of independent components. This approach separately addresses the influences of airfoil camber $m$ and a non-zero $\alpha _0$ on the mean flow as

(2.2) \begin{equation} L=L_0+\alpha _0L_{\alpha _0}+mL_m, \end{equation}

where $L_0$ represents the lift derived by Sears for a flat-plate airfoil at $\alpha _0=0$ , while $L_{\alpha _0}$ and $L_m$ denote the lift resulting from non-zero $\alpha _0$ and non-zero $m$ , respectively. In Atassi’s derivation, the amplitudes were expressed as $w_0=\tilde {k}_w^x/|\tilde {k}^x|\epsilon U_0$ and $u_0=\tilde {k}_w^x/|\tilde {k}^x|\epsilon U_0$ , where $\tilde {k}^{x}=\tilde {k}_w^x+\mathrm{i}\tilde {k}_u^x$ , $|\cdot |$ denotes the modulus of a complex number and $\epsilon U_0$ represents the total gust amplitude in the direction of propagation. This study exclusively examines airfoils with zero camber ( $m=0$ ), enabling the expression of normalized lift as

(2.3) \begin{equation} \frac {L}{\pi\rho c U_0 (\epsilon U_0)\mathrm{exp}(-\mathrm{i}\omega _w t)}=R_A\big(\tilde {k}_w^x,\tilde {k}_u^x,\alpha _0\big)=\frac {\tilde {k}_w^x}{\big|\tilde {k}^x\big|}\overline {S\big(\tilde {k}_w^x\big)}+\alpha _0R_{\alpha }\big(\tilde {k}_w^x,\tilde {k}_u^x\big), \end{equation}

where $R_A(\tilde {k}_w^x,\tilde {k}_u^x,\alpha _0)$ is the Atassi function, $\overline {(\cdot )}$ denotes the complex conjugate, $R_{\alpha }(\tilde {k}_w^x,\tilde {k}_u^x)$ is the response function of AoA.

It is important to note that the denominators of the first terms in (2.1) and (2.3) differ, resulting in the third term in (2.3) being expressed as ${\tilde {k}_w^x}/{|\tilde {k}^x|}\overline {S(\tilde {k}_w^x)}$ rather than $S(\tilde {k}_w^x)$ . For direct comparison with the Sears function in (2.1), (2.3) can be reformulated using the unified normalization as

(2.4) \begin{equation} \frac {L}{\pi\rho c U_0 w_0 \mathrm{exp}\!\left(\mathrm{i}\omega _wt\right)}=S\big( \tilde {k}_w^x\big)+\alpha _0\frac {\big|\tilde {k}^x\big|}{\tilde {k}_w^x}\overline {R_{\alpha }\big(\tilde {k}_w^x,\tilde {k}_u^x\big)}. \end{equation}

When $\alpha _0=0$ , (2.4) is equivalent to $S(\tilde {k}_w^x)$ . The disparity between the Sears and Atassi functions arises from the variation in $w_0$ and $\epsilon U_0$ . When $\tilde {k}_u^x$ equals zero, it follows that $\tilde {k}^x=\tilde {k}_w^x$ and $w_0=\tilde {k}_w^x/|\tilde {k}^x|\epsilon U_0=\epsilon U_0$ . Consequently, the Atassi problem simplifies to that of Sears. This highlights that in the case where AoA and camber are both zero, the difference between the theories of Sears and Atassi arises from the differing gust amplitudes associated with $\tilde {k}_u^x$ , as was first noted by Wei et al. (Reference Wei, Kissing, Wester, Wegt, Schiffmann, Jakirlic, Hölling, Peinke and Tropea2019). When the AoA is non-zero, the distinction is due to the influence of the airfoil on the velocity field in the case of Atassi, which has been partially confirmed experimentally by Young & Smyth (Reference Young and Smyth2021).

This study focuses on homogeneous and isotropic turbulence, with the turbulent flow generated by the uniform square grid considered nearly isotropic. Thus, when the gust propagates at $\tilde {k}_u^x=\tilde {k}_w^x$ , $\omega _w=\omega _u$ , (2.3) can be reformulated to express a single reduced frequency as follows:

(2.5) \begin{equation} R\big(\tilde {k}_w^x,\alpha _0\big)=\frac {\big|\tilde {k}^x\big|}{\tilde {k}_w^x}\overline {R_A\big(\tilde {k}_w^x,\alpha _0\big)}=S\big( \tilde {k}_w^x\big)+\alpha _0\frac {\big|\tilde {k}^x\big|}{\tilde {k}_w^x}\overline {R_{\alpha }\big(\tilde {k}_w^x\big)}. \end{equation}

To make comparisons easier, the Atassi function will be represented as $R(\tilde {k}_w^x,\alpha _0)$ in (2.5).

Combine (2.1) and (2.5) the generalized expression of lift on an airfoil is

(2.6) \begin{equation} {L=\pi \rho c U_0 \mathrm{exp}\!\left(\mathrm{i}\omega _w t\right)\chi\!\left(\widetilde {k}_w^x, \alpha _0\right)w_0}, \end{equation}

where $\chi (\widetilde {k}_w^x, \alpha _0)$ represents the one-wavenumber aerodynamic admittance. Whether to use the Sears or Atassi function is dependent upon the specific conditions.

Then the lift on an airfoil at $\alpha _0$ in a specified random velocity field can be constructed using Fourier superposition. Provided that the spanwise variation of the incident turbulent velocity is constant across the span (e.g. full spanwise correlation), the one-dimensional lift spectrum can be expressed as follows:

(2.7) \begin{equation} \phi _{L}\left({k}_w^x,\alpha _0\right)=\left(\pi\rho U_0c\right)^2\left|\chi\!\left(\widetilde {k}_w^x,\alpha _0\right)\right|^2\phi _w\left({k}_w^x\right), \end{equation}

where $|\chi (\widetilde {k}_w^x,\alpha _0)|^2$ represents the one-wavenumber aerodynamic admittance at an AoA of $\alpha _0$ , $\phi _w({k}_w^x)$ and $\phi _L({k}_w^x,\alpha _0)$ denote the one-dimensional spectra of vertical velocity and lift, respectively. When $\alpha _0$ is zero, $|\chi (\widetilde {k}_w^x,\alpha _0)|^2$ corresponds to the square of the modulus of the Sears function.

2.2. Methodology

This section outlines the difference between the one-wavenumber and generalized aerodynamic admittance. It also demonstrates the extended method used to calculate the generalized aerodynamic admittance at a non-zero (or zero) AoA in turbulence. The combined theoretical and experimental method of zero AoA has been explained in more detail as Yang et al. (Reference Yang, Li and Liao2019) and Li et al. (Reference Li, Zhao, Yang and Zhou2022). The experimental generalized admittance is conventionally derived from the measured spectrum of turbulence and the associated lift as

(2.8) \begin{equation} \left|A\left(\tilde {k}_w^x,\alpha _0\right)\right|^2=\frac {1}{\left(\pi\rho U_0 c\right)^2}\frac {\phi _L\left({k}_w^x,\alpha _0\right)}{\phi _w\left({k}_w^x\right)}. \end{equation}

In the context of three-dimensional turbulent flow, the assumption of full spanwise correlation fails. It is essential to introduce the spanwise wavenumbers $k_w^y$ and $k_u^y$ to characterize the spanwise variations. In isotropic turbulence, it can be assumed that $\omega _u=\omega _w$ , leading to the relationships $k_w^x=k_u^x$ and $k_w^y=k_u^y$ . The Fourier component is represented as a two-dimensional sinusoidal gust given by $w_G=w_0\mathrm{exp}[\mathrm{i}(\omega t-k_w^xx-k_w^yy)]$ and $u_G=u_0\mathrm{exp}[\mathrm{i}(\omega t-k_w^xz-k_w^yy)]$ , ( $w_0=u_0$ ), as illustrated in figure 1(c). According to the three-dimensional theory as in Yang et al. (2019), then the lift per unit span can be expressed as

(2.9) \begin{equation} {L=\pi\rho U_0 c \mathrm{exp}\!\left(\mathrm{i}\omega _w t\right)\mathrm{sinc}\!\left(k_w^y\Delta s/2\right)\chi\!\left(\tilde {k}_w^x, \tilde {k}_w^y, \alpha _0\right)w_0}, \end{equation}

where $\chi (\tilde {k}_w^x, \tilde {k}_w^y, \alpha _0)$ is the two-wavenumber aerodynamic admittance for a fixed $\alpha _0$ . In this study, the lift was obtained from pressure measuring strips, the sinc function $\mathrm{sinc}(k_w^y\Delta s/2)$ approaches unity, as the spanwise measurement length $\Delta s$ approaches zero. Hence, the two-dimensional lift spectrum for a fixed $\alpha _0$ in turbulence could be written as

(2.10) \begin{equation} \phi _{L}\left({k}_w^x,k_w^y,\alpha _0\right)=\left(\pi\rho U_0c\right)^2\left|\chi\!\left(\widetilde {k}_w^x,\widetilde {k}_w^y,\alpha _0\right)\right|^2\phi _w\big({k}_w^x,{k}_w^y\big), \end{equation}

where $\phi _w({k}_w^x,{k}_w^y)$ is the two-dimensional turbulence spectrum.

According to Li et al. (Reference Li, Yang, Li and Liao2018) and Yang et al. (2019), the two-wavenumber aerodynamic admittance $|\chi (\tilde {k}_w^x,\tilde {k}_w^y,\alpha _0)|^2$ could be expressed as the product of the one-wavenumber aerodynamic admittance $|\chi (\tilde {k}_w^x,\alpha _0)|^2$ and the spanwise influence term $\eta (\tilde {k}_w^x,\tilde {k}_w^y,\alpha _0)$ . The AoA influences both the one-wavenumber aerodynamic admittance $|\chi |^2$ and the spanwise influence term $\eta$ , as it changes the cross-sectional shape in relation to the streamwise direction. According to Yang et al. (Reference Yang, Li and Liao2019), the lift coherence function could be expressed as a function containing the spanwise influence term, based on the lift cross-spectrum and the inverse Fourier transform. Hence, the spanwise influence term can be experimentally determined by matching the measured spanwise coherence of lift with the theoretical expression as

(2.11) \begin{equation} Coh_L\big(k_w^x,\alpha _0,\Delta y\big)=\frac {\int _{-\infty }^{+\infty }\phi _{w}\left(k_w^x,k_w^y\right)\eta \left(\tilde {k}_w^x,\tilde {k}_w^y,\alpha _0\right)\cos \left(\mathrm{i}k_w^y\Delta y\right) \mathrm{d}k_w^y}{\int _{-\infty }^{+\infty }\phi _{w}(k_w^x,k_w^y)\eta (\tilde {k}_w^x,\tilde {k}_w^y,\alpha _0) \mathrm{d}k_w^y}. \end{equation}

Then the final one-dimensional lift spectrum is expressed by integrating out the spanwise wavenumber $k_w^y$ as follows:

(2.12) \begin{align} \phi _{L}\left(k_w^x,\alpha _0\right)&=\int _{-\infty }^{+\infty }\phi _{L}\left(k_w^x,k_w^y,\alpha _0\right) \mathrm{d}k_{w}^{y}\nonumber\\&=\left(\pi \rho U_0c\right)^{2}\left|\chi\!\left(\tilde {k}_w^x,\alpha _0\right)\right|^{2}g_{span}\left(\tilde {k}_w^x,\alpha _0\right)\phi _{w}\left(k_w^x\right), \end{align}
(2.13) \begin{align} g_{{span}}\left(\tilde {k}_w^x,\alpha _0\right)=\frac {\int _{-\infty }^{+\infty }\phi _{w}\left(k_w^x,k_w^y\right)\eta \left(\tilde {k}_w^x,\tilde {k}_w^y,\alpha _0\right) \mathrm{d}k_w^y}{\int _{-\infty }^{+\infty }\phi _{w}\left(k_w^x,k_w^y\right) \mathrm{d}k_w^y}, \end{align}
(2.14) \begin{align} \eta \left(\tilde {k}_w^x,\tilde {k}_w^y,\alpha _0\right)=\frac {1+a_1\pi\left(\tilde {k}_w^x\right)^{a_3}}{1+a_1\pi\left(\tilde {k}_w^x\right)^{a_3}+a_2\pi\left(\tilde {k}_w^y\right)^2}, \end{align}

where $g_{{span}}(\tilde {k}_w^x,\alpha _0)$ is a factor reflecting the effects of the spanwise influence term as in Li et al. (Reference Li, Zhao, Yang and Zhou2022). The free parameters $a_i (i=1,2,3)$ in the generalized expression of $\eta$ can be determined by matching. To determine the turbulent spectra $\phi _w(k_w^x,k_w^y)$ in (2.13), the von Kármán model was utilized as

(2.15) \begin{equation} \phi _w\left(k_w^x,k_w^y\right)=\frac {14.624\sigma ^2_w\left(L_w^x\right)^4\left[\left(k_w^x\right)^2+\left(k_w^y\right)^2\right]}{\left\{1+7.172\left(L_w^x\right)^2\left[\left(k_w^x\right)^2+\left(k_w^y\right)^2\right]\right\}^{7/3}}, \end{equation}

where $L_w^x$ is the vertical integral length scales in the streamwise direction, $\sigma _w$ represents the root mean square of vertical fluctuating velocity.

Substituting (2.15) into (2.13) yields a general expression for $g_{{span}}(\tilde {k}_w^x,\alpha _0)$ as follows:

(2.16) \begin{align} &g_{{span}}\left(\tilde {k}_w^x,\alpha _0\right)\nonumber\\&=\frac {\int _{-\infty }^{+\infty }\left[\left(\tilde {k}_w^x\right)^2+\left(\tilde {k}_w^y\right)^2\right]\eta \left(\tilde {k}_w^x,\tilde {k}_w^y,\alpha _0\right)\Bigg/{\left\{1+28.688\gamma ^2\left[\left(\tilde {k}_w^x\right)^2+\left(\tilde {k}_w^y\right)^2\right]\right\}^{7/3}} \mathrm{d}\tilde {k}_w^y}{\int _{-\infty }^{+\infty }\left[\left(\tilde {k}_w^x\right)^2+\left(\tilde {k}_w^y\right)^2\right]\Bigg/{\left\{1+28.688\gamma ^2\left[\left(\tilde {k}_w^x\right)^2+\left(\tilde {k}_w^y\right)^2\right]\right\}^{7/3}} \mathrm{d}\tilde {k}_w^y}, \end{align}

where $\gamma =L_w^x/c$ represents the scale ratio associated with the turbulent flow field. By combing (2.8) and (2.12), the generalized aerodynamic admittance can be calculated as

(2.17) \begin{equation} \left|A\left(\tilde {k}_w^x,\alpha _0\right)\right|^2=\frac {\phi _L\left({k}_w^x,\alpha _0\right)}{\left(\pi\rho U_0 c\right)^2\phi _w\left({k}_w^x\right)}={g_{{span}}\left(\tilde {k}_w^x,\alpha _0\right)}\left|\chi\!\left(\widetilde {k}_w^x,\alpha _0\right)\right|^2. \end{equation}

Apparently, the generalized aerodynamic admittance is influenced by both the variation of the one-wavenumber aerodynamic admittance and the spanwise influence term. The theories reviewed above indicate that the one-wavenumber admittance may correspond to either the Sears or Atassi functions. Determining the spanwise influence term would enable the calculation of the generalized admittance at the corresponding AoA. This study will examine the impact of AoA on aerodynamic admittance of an airfoil prior to prestall, based on the extended methodology.

3. Experimental set-up

The experiments and results presented in this work were carried out in the XNJD-1 closed-loop wind tunnel at Southwest Jiaotong University, with an overview in the previous study (Zhao et al. Reference Zhao, Yang, Li and Peng2023). The cross-sectional area of the test section measures $2.40\times 2.00$ $\mathrm{m}^2$ . The background turbulence intensity is lower than 0.5 %. A uniform square biplane grid with a mesh size of $M=450$ mm and a bar size of $d=85$ mm was placed at the front of the test section to generate turbulence. Instantaneous velocities were measured using TFI Cobra Probe anemometers, which have an accuracy of less than 0.5 m $\mathrm{s^{-1}}$ and can detect the turbulence intensity of approximately 30 %. Each measurement consisted of a data sampling frequency of 1024 Hz and the sampling time was set as 120 s, with five measurements conducted at each position to minimize statistical errors. The distance from the measurement point to the grid was established at approximately $x_m=5.75$ m, and the mean flow velocity was set as 10.5 m $\mathrm{s^{-1}}$ .

Figure 2.

Schematic of the experimental set-up of pressure measurement on a fixed NACA 0015 airfoil model in the turbulent flow: (a) side view; (b) top view.

A NACA 0015 airfoil, with a chord length of $c$ = 0.4 m and a span of $s$ = 1.2 m, was constructed and used as the pressure measurement model in this investigation, as shown in figure 2. End plates were installed on both sides of the model to prevent airflow around the tips. Two strips, each with 64 taps, were placed on the central section of the airfoil surface. The spanwise distance between the two pressure measuring strips is $\Delta y$ = 0.05 m. All taps were connected to two MPS4164, 64-channel electronic pressure scanning modules, employing the same digital service module, Scanivalue DSM4000. The Scanivalve MPS4164 has a full scale pressure range of 10.16 cm of water column, with an uncertainty of $\pm 0.08\,\%$ full scale. All tubes linking the tap and scanner were of consistent length, limited to 0.2 m, to reduce the influence of resonance and damping effects on the measured signal to a level that rendered them negligible. In the experiments, the sampling frequency of pressure measurement was 256 Hz and the pressure measuring sampling time was set as 120 s, which was the same as that of velocities measurement. The total lift force is calculated by integration as $L=\sum (p_i-p_0)\delta _i \mathrm{cos}\theta _{i,\alpha _0}$ , where $\delta _i$ is the integration length of the tap $i$ , and $\theta _{i,\alpha _0}$ indicates the angle between the tangent line at the measurement point $i$ and the inner normal. It is important to recognize that $\theta _{i,\alpha _0}$ depends on both $i$ and $\alpha _0$ .

4. Results and discussion

4.1. Characterization of turbulent flow

Figure 3.

(a) Spanwise distributions of streamwise and vertical turbulence intensity. (b) One-wavenumber spectra of turbulent streamwise and vertical velocity matched with theoretical models.

The homogeneity of the turbulent flow field generated by the grid is evaluated first at the measurement location. The statistical properties of homogeneous turbulence are invariant in spatial locations; therefore, the root mean square of the fluctuating velocity $\sigma _i (i=u,v,w)$ within a given cross-section should remain uniform. Figure 3(a) presents the spanwise distribution of $\sigma _u$ and $\sigma _w$ . The shadow range refers to the spanwise length of the airfoil model. The normalized turbulent intensities at different spanwise positions are nearly identical, suggesting a significant level of homogeneity within the turbulent flow field generated in this study.

Figure 3(b) shows that the measured turbulent spectrum agrees well with the von Kármán spectral model. This model allows for the determination of the vertical integral length scale in the streamwise direction, as $L_w^x=0.140$ m. The value of the vertical integral length scale in the spanwise direction $L_w^y$ is 0.132, which can be calculated based on its definition. In this study, the value of the scale ratio $\gamma =L_w^x/c$ in (2.16) is calculated to be 0.35. In the context of homogeneous and isotropic turbulence, $L_w^x/L_w^y=1$ is expected as indicated by Kitamura et al. (Reference Kitamura, Nagata, Sakai, Sasoh, Terashima, Saito and Harasaki2014). The experimental result indicates that $ L_w^x/L_w^y = 1.06$ . Achieving fully isotropy is challenging; however, it may be regarded as approximately isotropic. Hence, the theoretical two-dimensional turbulence spectra, as shown in (2.15), could be calculated using these characteristics.

4.2. Lift coefficients and coherence at various AoAs

Figure 4.

Comparison of measured lift coefficients in turbulence with thin airfoil theory and experimental values for various Reynolds numbers.

Figure 5.

Comparisons between the direct measurement results of the spanwise coherence of lift and the fitted values at various AoAs, accompanied by coherence function curves calculated based on approximate expressions of theoretical spanwise influence terms: (a) $\alpha _0=0^{\circ }$ ; (b) $\alpha _0=2^{\circ }$ ; (c) $\alpha _0=4^{\circ }$ ; (d) $\alpha _0=6^{\circ }$ ; (e) $\alpha _0=8^{\circ }$ ; (f) $\alpha _0=10^{\circ }$ .

In § 2, we know that the stall must be avoided to satisfy the requirements of the applications. Figure 4 compares the measured lift coefficient in turbulent flow with the prediction of thin airfoil theory, as well as the experimental results of Sheldahl & Klimas (Reference Sheldahl and Klimas1981) for various Reynolds numbers $Re$ . This result not only indicates at what AoA the airfoil in turbulence will begin to stall but also verifies the reliability of the pressure measurement experiment. According to figure 4, the results of AoA from $0^{\circ }$ to $10^{\circ }$ (with an interval of $2^{\circ }$ ) will be discussed.

To address the issue of spanwise effects, the spanwise influence term $\eta (\tilde {k}_w^x,\tilde {k}_w^y,\alpha _0)$ in (2.14) should be determined first. Using (2.11) to compare with the measured lift coherence allows experimental determination of the spanwise influence term $\eta$ . Besides, another approximate expression of the spanwise influence term will also be substituted into (2.11) for calculations. The exact function based on thin-plate theory was numerically computed by Graham (Reference Graham1970, Reference Graham1971) and the approximate expression of the spanwise influence term was derived by Mugridge (Reference Mugridge1971) as

(4.1) \begin{equation} \eta _{\mathrm{TP}}\left(\tilde {k}_w^x,\tilde {k}_w^y\right)=\frac {\left(\tilde {k}_w^x\right)^{2}+{2}/{\pi^2}}{\left(\tilde {k}_w^x\right)^{2}+\left(\tilde {k}_w^y\right)^{2}+2/\pi^2}. \end{equation}

Figure 5 presents the measured values of lift coherence before prestall with the theoretical and fitted results. The theoretical values are obtained by substituting $\eta _{\mathrm{TP}}$ from (4.1) into (2.11), respectively. From the results of the lift coherence, it can be seen that the approximate expression of Mugridge (Reference Mugridge1971) performs well at low reduced frequencies and overestimates it at high frequencies. This could be because the effects of thickness are not taken into account in thin-plate theory. However, the empirical expression of $\eta$ in (2.14) includes three adjustable parameters to enable a good fitting to the experimental results, resulting in a more accurate spanwise influence term. In addition to the spanwise effect, the effects of three-dimensionality also includes the effect of distortion. According to Goldstein (Reference Goldstein1978) and Hunt & Carruthers (Reference Hunt and Carruthers1990), in three-dimensional turbulence, the mean flow field around the airfoil stretches and rotates the vorticity. This effect might occur with the increase of airfoil AoA, but it does not exist in the two-dimensional flow model. It is of the same order as the AoA effect in form, but appears to be smaller in magnitude. It can be seen that with an increase in AoA, the lift coherence exhibits a slight change; however, this change is not substantial, leading to the fact that the lift coherence of the fitted result remains largely unchanged.

4.3. Spanwise effects at different AoAs

Figure 6.

(a) The spanwise influence term based on thin-plate theory. (b) The spanwise influence term at zero AoA determined from lift coherence.

After obtaining the spanwise influence term determined from lift coherence, figure 6 compares the values at zero AoA with the approximate expression $\eta _{\mathrm{TP}}$ based on thin-plate theory. It can be seen that when the reduced frequency is relatively small, the results obtained through the lift coherence are almost identical to the approximate values of the thin-plate theory. However, as $k_w^xc/2$ increases, the experimental results in figure 6(b) are significantly smaller than the theoretical approximate expression in figure 6(a), and this difference also increases with the increase of $k_w^xc/2$ . This could be due to the effect of thickness becoming increasingly apparent as the reduced frequency increases when AoA is zero. Then the variation of the free parameters in the spanwise influence term with AoA inevitably influences $g_{{span}}$ in (2.16). As in (2.17), the generalized aerodynamic admittances could be considered the product of one-wavenumber aerodynamic admittance and $g_{{span}}$ . Once $g_{{span}}$ is determined, the corresponding generalized aerodynamic admittances can also be obtained.

Figure 7 describes the variation of $g_{{span}}$ with AoA relative to the scale ratio in this study. It can be seen that $g_{{span}}$ based on $\eta _{\mathrm{TP}}$ is consistent with the experimental results at low frequencies, but differences also appear as the reduced frequency increases. This is similar to the results in figures 5 and 6 that the results of thin-plate theory and experimental results are almost the same at low frequencies, whereas the larger difference appears with increasing frequencies. It is evident that the results from the approximate expression based on the thin-plate could not fully represent the actual spanwise effects, whether due to differences in the thickness of the actual airfoil or the approximation method, even at zero AoA. Figure 7 also indicates that as AoA increases, $g_{{span}}$ exhibits a slight increase; however, a notable increase occurs from $8^{\circ }$ to $10^{\circ }$ . This may be attributed to the near prestall condition at an AoA of $10^{\circ }$ . Furthermore, the variation of $g_{{span}}$ with AoA contributes to the further variation of the generalized aerodynamic admittance with AoA.

Figure 7.

Curves of the spanwise effect factors corresponding to different AoAs based on fitted spanwise influence term and the approximate expression based on thin-plate theory.

4.4. One-wavenumber and generalized aerodynamic admittances

Figure 8.

(a) One-wavenumber aerodynamic admittances based on the Sears function and Atassi function at various AoAs. (b) Calculated generalized aerodynamic admittances based on experimental fitted results at various AoAs.

Figure 8(a) illustrates the one-wavenumber aerodynamic admittances, computed using the Sears and Atassi functions, under the assumption of full correlation in the spanwise direction. Figure 8(b) presents the generalized aerodynamic admittance derived from the fitting results. Figure 8(a) demonstrates that after unified normalization, the Sears function is identical to the Atassi function at an AoA of zero. With the emergence of non-zero AoA, the Atassi functions with different $\alpha _0$ are required to characterize the corresponding one-wavenumber aerodynamic admittance. With an increase in AoA, the one-wavenumber aerodynamic admittance also rises at low frequencies, surpassing the quasisteady value of the Sears function, which is 1.0. This could be attributed to the fact that the streamwise turbulent velocities also contribute to the lift of an airfoil with non-zero AoA. To align with the Sears function, only the vertical component was used to normalize, leading to an increase in one-wavenumber aerodynamic admittance with increasing AoA at low frequencies. As the frequency increases, the one-wavenumber aerodynamic admittance at various AoA tends towards the same value.

Figure 9.

Measured generalized aerodynamic admittances compared with the calculated values at various AoAs: (a) $\alpha _0=0^{\circ }$ ; (b) $\alpha _0=2^{\circ }$ ; (c) $\alpha _0=4^{\circ }$ ; (d) $\alpha _0=6^{\circ }$ ; (e) $\alpha _0=8^{\circ }$ ; (f) $\alpha _0=10^{\circ }$ .

After evaluating the influence of spanwise effects as shown in figure 5, the generalized aerodynamic admittance could be obtained as (2.17). The calculated generalized aerodynamic admittances as depicted in figure 8(b) show a notable reduction in comparison with the one-wavenumber aerodynamic admittance. Taking into account $g_{{span}}$ as shown in figure 7, the aerodynamic admittances experienced a significant compression, leading to a reduction in the influence of AoA. Figure 7 indicates that as AoA increases, $g_{{span}}$ exhibits a slight increase; however, a notable increase occurs from $8^{\circ }$ to $10^{\circ }$ . This may be attributed to the near prestall condition at an AoA of $10^{\circ }$ . This leads to a more notable increase in generalized aerodynamic admittance when AoA is transitioning from $\alpha _0=8^{\circ }$ to $\alpha _0=10^{\circ }$ .The differences induced by AoA were diminished in absolute values in generalized aerodynamic admittance. The higher AoA correlates with increased $g_{{span}}$ , leading to accelerated growth of generalized aerodynamic admittance as the AoA increases.

Figure 9 compares the measured generalized aerodynamic admittance with the calculated values obtained through various methods to assess the precision of various calculated aerodynamic admittance in representing the generalized aerodynamic admittance of an airfoil at different AoA in the actual turbulent flow. In figure 9, the results obtained directly from the Sears and Atassi functions is one-wavenumber aerodynamic admittance, which is the theoretical results without considering the spanwise influence term, corresponding to the curves in figure 8(a). Calculations I and II, i.e. calculated generalized aerodynamic admittance as shown in figure 8(b) and (2.17), are the results of taking $g_{{span}}$ into account. The difference between them is that Calculation I is based on the Sears function, while the one-wavenumber aerodynamic admittance used in Calculation II is the Atassi function at the corresponding AoA.

It can be seen that after considering the spanwise effects, the Sears function and the Atassi function collapse into Calculations I and II, respectively. Compared with the measured values, one-wavenumber aerodynamic admittance significantly overestimates the measured aerodynamic admittance in turbulent flow, probably due to the assumption of complete correlation in theory. The introduction of $g_{{span}}$ corrected the difference between the assumption and actual situation through matching the lift spanwise coherence, thus calculating the corresponding generalized aerodynamic admittance. Obviously, it can be found that Calculation I agrees well with the measured values when AoA equals zero. However, as AoA increases, the measured generalized aerodynamic admittance gradually deviates from Calculation I, indicating that AoA does have an impact on the generalized aerodynamic admittance in turbulent flow. Before prestall, the larger the AoA, the higher the generalized aerodynamic admittance at low frequencies., In addition, the growth rate of the generalized aerodynamic admittance also increases with larger AoA. However, Calculation II shows good agreement for the generalized aerodynamic admittance at different AoAs in turbulent flow, indicating the effective application of the Atassi function in turbulence after considering the spanwise effects.

5. Conclusions

This research investigates the lift aerodynamic admittances of a NACA 0015 airfoil at different AoAs in turbulence. A methodology that combines theoretical and experimental approaches is extended to derive the generalized aerodynamic admittances of an airfoil at various AoAs. For an airfoil at an AoA of zero, the Atassi and Sears functions are identical after unified normalization. For non-zero AoA, the influence of streamwise fluctuating wind speed on lift is unavoidable. Consequently, after the normalization that is the same as the Sears function, the one-wavenumber aerodynamic admittances derived from the Atassi function for a non-zero AoA will be larger than that of the Atassi function at a zero AoA and the Sears function. Although the one-wavenumber aerodynamic admittances derived from the Atassi function represents the promoting influence of AoA on aerodynamic admittances, it significantly overestimates the overall values possibly due to the assumption of complete correlation becoming invalid in turbulence. Analysis of the experimental lift coherence reveals that prior to prestall conditions, the factor of spanwise influence term increases with the AoA, and the growth rate similarly grows with the AoA. The introduction of the modified factors of the spanwise influence term results in the collapse of the one-wavenumber aerodynamic admittances, based on the Atassi function, into the generalized aerodynamic admittances. The generalized aerodynamic admittances, derived from the effects of AoA both on the spanwise influence term and the one-wavenumber aerodynamic admittances from the Atassi function, effectively assesses the enhancement influence of larger AoA before stall on aerodynamic admittances in turbulent flow.

Acknowledgements.

This work also benefits from discussions with Nathaniel J. Wei about the Atassi function. Lastly, the authors would like to express the gratitude to the reviewers from the Journal of Fluid Mechanics for their detailed comments and helpful suggestions that significantly improved the arguments and quality of the paper.

Funding.

This work is supported by the National Natural Science Foundation of China under grant nos. 52008357 and 52408550 and the Natural Science Foundation of Sichuan Province under grant no. 2025ZNSFSC0405.

Declaration of interests.

The authors report no conflict of interest.

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Figure 0

Figure 1.(a) An airfoil at an AoA encountering a sinusoidal vertical gust. (b) An airfoil at an AoA encountering a sinusoidal streamwise gust. (c) Schematic of the problem: an airfoil at an AoA encountering the combined Fourier components of streamwise and vertical fluctuating velocities with spanwise variation.

Figure 1

Figure 2.Schematic of the experimental set-up of pressure measurement on a fixed NACA 0015 airfoil model in the turbulent flow: (a) side view; (b) top view.

Figure 2

Figure 3.(a) Spanwise distributions of streamwise and vertical turbulence intensity. (b) One-wavenumber spectra of turbulent streamwise and vertical velocity matched with theoretical models.

Figure 3

Figure 4.Comparison of measured lift coefficients in turbulence with thin airfoil theory and experimental values for various Reynolds numbers.

Figure 4

Figure 5.Comparisons between the direct measurement results of the spanwise coherence of lift and the fitted values at various AoAs, accompanied by coherence function curves calculated based on approximate expressions of theoretical spanwise influence terms: (a) $\alpha _0=0^{\circ }$; (b) $\alpha _0=2^{\circ }$; (c) $\alpha _0=4^{\circ }$; (d) $\alpha _0=6^{\circ }$; (e) $\alpha _0=8^{\circ }$; (f) $\alpha _0=10^{\circ }$.

Figure 5

Figure 6.(a) The spanwise influence term based on thin-plate theory. (b) The spanwise influence term at zero AoA determined from lift coherence.

Figure 6

Figure 7.Curves of the spanwise effect factors corresponding to different AoAs based on fitted spanwise influence term and the approximate expression based on thin-plate theory.

Figure 7

Figure 8.(a) One-wavenumber aerodynamic admittances based on the Sears function and Atassi function at various AoAs. (b) Calculated generalized aerodynamic admittances based on experimental fitted results at various AoAs.

Figure 8

Figure 9.Measured generalized aerodynamic admittances compared with the calculated values at various AoAs: (a) $\alpha _0=0^{\circ }$; (b) $\alpha _0=2^{\circ }$; (c) $\alpha _0=4^{\circ }$; (d) $\alpha _0=6^{\circ }$; (e) $\alpha _0=8^{\circ }$; (f) $\alpha _0=10^{\circ }$.