0 Introduction

Tonal noise emitted from an isolated airfoil at low and moderate Reynolds numbers has been identified as one of the most remarkable problems both in academic and engineering fields^[1], especially for the small wind turbines and micro air vehicles. Despite the fact that many investigations have been conducted experimentally and numerically to clarify the physics behind it, the mechanism of the tonal noise generation has not yet been clearly understood. The first comprehensive study on the tonal noise from an airfoil was conducted by Paterson et al.^[2], who did experimental investigations of NACA0012 and NACA0018 airfoils at a wide range of Reynolds numbers and angles of attack. They stated that the generation of the tonal noise was associated with a vortex-shedding mechanism near the trailing edge. Tam^[3] argued that the hypothesis of the vortex- shedding mechanism itself was not valid. Instead, he proposed a self-excited feedback loop mechanism to explain the generation of the discrete tones. Arbey and Bataille^[4] studied a NACA0012 airfoil at zero angle of attack experimentally, observing that the acoustic spectrum actually contains both broadband components with a peak frequency and discrete components in form of equal-spaced discrete frequencies. They owed the broadband components to the diffraction of Tollmien-Schlichting (T-S) waves at the sharp trailing edge of the airfoil and predicted the discrete components via Tam^[3]'s feedback loop model. Lowson, Fiddes, and Nash^[5] noticed that the generation of tonal noise was related to the laminar separation on the pressure side of the airfoil. Nash, Lowson, and Mcalpine^[6] and Mcalpine, Nash, and Lowson^[7] later investigated a NACA0012 airfoil at small angles of attack in an anechoic wind tunnel. They proposed a revised feedback loop model based on the amplification of T-S waves in the laminar separation region on the pressure side of the airfoil. Considering the two-dimensional nature of the tonal noise generation, Desquesnes, Terracol, and Sagaut^[8] conducted two-dimensional direct numerical simulations(DNS) and observed similar acoustic spectra to those measured by Arbey and Bataille^[4]. In their efforts to explain the multiple tones in the spectra, they suggested that a secondary feedback loop is also existed on suction side of the airfoil, which serves to modulate the amplitude of the main feedback loop on the pressure side of the airfoil. Using DNS, Jones and Sandberg^[9] observed that the most amplified frequency of the T-S waves did not approximate to the primary tone frequency. They proposed a mechanism involving global instabilities around the whole airfoil to account for the tonal noise generation. Tam and Ju^[10] numerically examined the sound generation from NACA0012 at zero angle of attack and found only a single peak in the acoustic spectrum. They ascribed this single tone to the Kelvin-Helmholtz (K-H) instabilities in the wake rather than the T-S waves in the upstream boundary layers. Recently, active and passive flow control methods^[11-12] have been successfully implemented to suppress the generation of the tonal noise by interfering the growth of T-S waves within the boundary layers, which to some extent indicate the pivotal role of boundary layer instabilities for the noise generation.

Aforementioned studies were performed at moderate Reynolds numbers (Re_c=5×10⁴—5×10⁵). Apart from these studies, Ikeda, Atobe, and Takagi^[13] numerically investigated NACA0012 and NACA0006 airfoils at a low Reynolds number (Re_c=1×10⁴). They concluded that at zero angle of attack wake instabilities dominate the tonal noise generation. As the angle of attack and freestream Mach number increase, T-S waves are enhanced and a feedback loop is gradually established to determine the tone frequencies. The underlying mechanism of the tonal noise generation seems to differ regarding different flow conditions, such as Reynolds numbers, angles of attack, and freestream Mach numbers.

The objective of the present study is two-fold: to put more insight into the tonal noise generation from an airfoil at both low and moderate Reynolds numbers, and to further examine the role of boundary layer instability on the suction and pressure side of the airfoil.Section Ⅱ gives an introduction of the numerical methods. Section Ⅲ presents analysis on acoustic fields and tonal noise generation mechanisms. Finally, concluding remarks are summarized in Section Ⅳ.

1 Numerical methods 1.1 Flow conditions

Flow past a symmetric airfoil NACA0015 with a sharp trailing edge is simulated. The freestream Mach number is M=0.2, and the Reynolds number based on the chord lengthC is chosen to be Re_c=1×10⁴ for low Reynolds number cases and Re_c=5×10⁴ for moderate Reynolds number cases. Three angles of attack, 0°, 2° and 4°, are investigated. Details of the flow conditions are listed in Table 1. For simplicity, the simulation cases are labeled using the Reynolds number and the angle of attack. For instance, R1A0 represents the case operating at Reynolds number of Re_c=1×10⁴ and zero angle of attack.

1.2 Numerical algorithm

The governing equations are two-dimensional unsteady compressible Navier-Stokes equations. A finite difference approach is used to solve the governing equations. In order to meet low-dispersive and low-dissipative requirements of turbulent flows and aerodynamic sound, the convective terms are discretized by means of a six-order compact differencing scheme in transformed curvilinear coordinates^[14-15]. An tenth-order low-pass spatial filtering scheme (with optimization parameter α_f=0.494^[14]) is applied at each time step on the conservative variables to ensure numerical stability. Viscous terms are evaluated by a sixth-order central differencing scheme. Alternate directional implicit symmetric Gauss-Seidel (ADI-SGS) scheme^[16] is performed for the time integration. A second-order temporal accuracy is obtained with three local subiterations to converge the Newton-Raphson subiterative process.

The computational grids are shown in Figure 1. The grids are designed for Re_c=5×10⁴ cases and used for the low Reynolds number cases as well. Structured grids with a C-type grid-topology are adopted. The outer boundaries are about 100 times of the chord length away from the airfoil surface. Grids are divided into an upstream block from trailing edge and a wake block behind it. The two blocks are connected with 12 overlap nodes. Details about the grid points are listed in Table 2. The minimum grid sizes in wall-normal and streamwise direction, normalized by the chord length c, are 1×10^-4 and 2×10^-3 respectively, which ensure all grids satisfy the inequalities^[17]: Δξ⁺ < 30, Δη⁺ < 1, where Δ denotes the minimum grid spacing and + denotes a normalization based on viscous unit. Non-slip and adiabatic conditions are applied on the wall surface. Zero-gradient pressure condition is employed at the outlet boundaries.

The time step non-dimensonalized by the chord length and the freestream velocity is 4×10^-5, with the maximum CFL number less than unit. The total time steps calculated are 6×10⁵, corresponding to a time duration that the freestream flow pass a distance of 24 chord length. After the computation of the first 2×10⁵ time steps, flow reaches a quasisteady state, then instantaneous flow fields of the remaining 4×10⁵ time steps are collected for obtaining time-averaged and statistical data for subsequent analysis.

1.3 Validations

The CFD codes in use have been extensively validated by several applications, such as cavity noise and flow over single-element airfoils^[17-22]. In order to ensure the validity of the current calculations, the time-averaged pressure coefficients of the R5A2 and R5A4 case are compared with the experimental data given by Asada et al.^[23], as displayed in Figure 2. The experiments were conducted at either Re_c=4.4×10⁴ or Re_c=6.3×10⁴. It can be seen that the simulation results have acceptable agreements with the experimental results, confirming the validity of current study. Furthermore, it is worthy to mention that Lee et al.^[22] (using the same CFD codes) has demonstrated that two-dimensional DNSs performed at Reynolds numbers between 1×10⁴ and 5×10⁴ can predict the aerodynamic characteristics of the airfoil qualitatively. Therefore, the two-dimensional computations are considered as an appropriate approach within the current range of Reynolds numbers.

2 Results and discussions 2.1 Acoustic fields

The aerodynamic sound fields scattered from the trailing edge are examined. Due to the singularity of velocity in an inviscid limit, periodic vortical motions develop in the wake of the airfoil and very large vorticity variations are induced near the trailing edge^[13]. Contours of the divergence of instantaneous velocity ∇·u for the six cases are given in Figure 3. A general view is that all acoustic fields display typical bipolar behaviors, and acoustic waves are emitted from the trailing edge, which are consistent with the trailing edge noise features^{[8, 10, 13]}. The acoustic wavelengths are smaller in the moderate Reynolds number cases than those in the low Reynolds number cases, indicating higher dominant frequencies at the moderate Reynolds number.

Power spectral density (PSD) of the acoustic signals are obtained using time histories of pressure fluctuations sampled at a station point, where locates vertically upward from the trailing edge with a distance of 3c. The sample location is chosen to avoid contamination of the acoustic signals caused by hydrodynamic fluctuations^[8]. Each sample set, containing 4×10⁵ pressure fluctuation points, is divided into 8 segments with 50% overlaps, and each segment is estimated with a periodogram method using a Hamming window function. Then the estimations are averaged to provide the final PSD result of the sample set.

Figure 4 shows the PSD results, in which the frequency f^* is normalized by the chord length and the freestream velocity. The amplitude is normalized by its maximum value. It can be seen that for the low Reynolds number cases, only one single tone dominates the acoustic spectrum. As the angle of attack increases, the single tone displays a higher-frequency harmonic for the R1A4 case. For the moderate Reynolds number cases, discrete frequencies imposed on humps are identified, similar to the observation of Arbey and Bataille^[4] and Desquesnes, Terracol, and Sagaut^[8]. At zero angle of attack case (R5A0) there is only one hump in the spectrum. For the other two cases (R5A2 and R5A4), two humps are observed. The second hump is illustrated with a dashed rectangular frame. After conducting a careful inspection, it confirms that the second hump is not the higher-frequency harmonic of the first one. The two humps in the spectrum are related to different noise sources, which will be discussed in following sections. The tonal frequencies f_n as well as the primary frequency f_n^* in the spectra of the six cases are listed in Table 3. The primary frequency f_n^* in the moderate Reynolds number cases equals f_n which has the highest power spectral energy ineach hump, following Arbey and Bataille's^[4] approach.

2.2 Tonal noise generation

In this subsection, tonal noise generation from the trailing edge is investigated. Since a single tone dominates the spectrum in low Reynolds number cases, the instantaneous flow fields can be averaged into a phase-locked period using phase-averaging analysis^[19]. All snapshots of the instantaneous flow fields are reassembled into one typical cycle with 20 phases to filter out irregular small disturbances. Similar features are exhibited in the three cases operating at the low Reynolds number, so only the R1A2 case is used to discuss the tonal noise generation.

Figure 5 displays one typical loop of noise generation using phase-averaged flow fields for the R1A2 case, illustrated by variations of streamwise vorticity (a₁-a₄), Q criterion (b₁-b₄), and pressure fluctuations (c₁-c₄) at four different phases around the airfoil and its trailing edge. At ϕ=π/2, it displays that a laminar separation without reattachment occurs in the boundary layer on the suction side of the airfoil, while the boundary layer on the pressure side of the airfoil remains attached. A vortex V_p has just passed over the airfoil, and a new vortex V_n is generating near the trailing edge. Low pressure region is associated with the core of the vortex V_p, and the pressure gradient is balanced with the centrifugal force of the vortex^[19]. At ϕ=π, the scale of the V_n grows due to Kelvin-Helmholtz instability. The scattering of the vortex V_p occurs near the trailing edge, accompanying by velocity distortions and deformations. The balance between the pressure gradient and the centrifugal force of the vortex is broken up. The low pressure region is no longer associated with the vortex and begins to spread to the upper side of the trailing edge. High pressure values in the lower side of the trailing edge are generated because of the stagnation between two neighbored vortices. At ϕ=3π/2, the vortex V_n convects downstream, and the acoustic wave propagates to a far field. A new vortex V′_p is formed on the suction side of the airfoil. At ϕ=2π, the passage of the vortex V′_p leads to another half period of pressure oscillations near the trailing edge. The pressure oscillations near the trailing edge and the generation of the acoustic waves in the far fields are exactly related to the successive passage of vortices.

For moderate Reynolds number cases, it is difficult to conduct phase-averaged analysis because of the multiple tones. To confirm that the tones are linked to the vortex shedding as well as to clarify different roles of the vortex shedding on the suction and pressure side of the airfoil, PSD of an integral vorticity Ω along a perpendicular line away from the airfoil surface at x/c=0.93 are calculated. The definition of Ω is given by:

$ \mathit{\Omega } = \int_{s = 0}^{s = 0.2c} {\left| {\omega {'_x}} \right|} {\rm{d}}s $

(1)

where ω_x^' is local streamwise vorticity fluctuations, s is the coordinate of the line away from the airfoil surface. The length of 0.2c is selected to ensure the boundary layers are included. The peak frequencies in the PSD of Ω are denoted by f_Ω, which are in Table 4. It finds that f_Ω agree well with f_n listed in Table 3 for all simulated cases. Especially for the R5A2 and R5A4 case, f_Ω on the suction side and the pressure side of the airfoil are not identical but respectively equal f_n in the second and first hump in the acoustic spectra as displayed in Figure 4. It further confirms that the vortex shedding on the suction side and the pressure side of the airfoil contributes separately to the generation of sound waves. Above analysis demonstrates that the unsteady vorticity variations are accompanied with periodic pressure oscillations near the trailing edge which are observed as discrete tonal noise in the far field.

2.3 Linear stability analysis

To investigate the link between hydrodynamic stabilities and vortex formation in boundary layers, linear stability analysis (LSA) is performed by solving a spatial Orr-Sommerfeld equation (OSE) to model T-S waves in the boundary layers. The spatial OSE is given as follows:

$ \begin{array}{l}{(\overline{U} \alpha-\omega)\left(\phi^{\prime \prime}-\alpha^{2} \phi\right)-\overline{U^{\prime \prime}} \alpha \phi} \\ {\qquad=-\frac{1}{i R e}\left(\phi^{\prime \prime \prime\prime}-2 \alpha^{2} \phi^{\prime \prime}+\alpha^{4}\right)}\end{array} $

(2)

where U is the local time-averaged velocity profiles at stations perpendicular to the airfoil surface, α and ω=2πf are respectively the complex wave number and the angular frequency of T-S waves, ϕ is the non-dimensional amplitude of the disturbances (or instability waves). When a frequency is selected, the image part of α solved from the OSE, α_i^*, represents the growth rate of the T-S waves at such frequency.

Local growth rates of T-S waves on both sides of the airfoil for the low Reynolds number cases are investigated. The stations are selected every 10% chord from the leading edge to the trailing edge of the airfoil. Since no amplification of T-S waves is observed on the pressure side of airfoil in the R1A2 and R1A4 case, the LSA results only on the suction side of the airfoil are plotted in Figure 6. In the R1A0 case T-S waves emerge approximately at station of x/c=0.5, and the growth rates are amplified in the downstream, reaching a peak near the trailing edge. As the angle of attack increases, the locations where the T-S waves first detected move upstream, and the maximum growth rate of T-S waves is increased.

Local growth rates of T-S waves for the moderate Reynolds number cases are plotted in Figure 7 and 8, respectively obtained from the suction and pressure side of the airfoil. It is observed that on the suction side the growth rates of T-S waves first increase and then decrease along the streamwsie direction. This is caused by the reattachment of the boundary layer on the suction side of the airfoil, which leads to a relatively stable state again, as discussed in our previous work^[24]. For the R5A2 and R5A4 case, amplification of T-S waves is also identified on the pressure side, as shown in Figure 8. The growth rates remain increasing till the trailing edge, similar to the observation of low Reynolds number cases.

The total spatial growth rate of the T-S waves at a certain frequency are estimated by integration of the local spatial growth rates at each station. It is denoted as N-factors calculated by $ N=\ln \left(\int_{x_{\mathrm{s}}}^{x_{\mathrm{e}}} \alpha_{i}^{*} \mathrm{d} x\right)$, where x_s and x_e represent the starting and ending location, respectively. The most unstable frequency of the T-S waves in the boundary layer can be obtained with the N-factors, as shown in Figure 9 and 10. Vertical lines represent the primary frequencies of the vortex shedding (f_Ω^*) detected from the PSD of Ω with maximum energy. It can be seen that the primary frequencies of the vortex shedding on both sides of airfoil are all in vicinities of the most unstable boundary layer instabilities. Despite some minor deviations, considering that the parallel flow assumptions are not strictly meet here, it is possible to conclude that the formation of the vortices and the subsequent vortex shedding are associated with the spatial growth of the most unstable T-S waves.

2.4 Feedback loop mechanism and mathematical modeling

It has been clear that for low Reynolds number cases, the most unstable T-S waves give rise to the vortex formation and shedding at the same frequency of the unstable T-S waves. Vortex scattering near the trailing edge results in the generation of acoustic waves at that specific frequency, in terms of a single tone observed in the far field. The generation of the single tone in the low Reynolds number cases is probably associated with the inherent hydro-dynamic (local) instability^[25] of the boundary layer rather than a global instability evolving acoustic excitation or interaction. It seems to be contrary to Ikeda, Atobe, and Takagi's^[13] conclusion that a feedback loop mechanism characterizes the low Reynolds number airfoils.

For the moderate Reynolds number cases, the multiple tones in the acoustic spectra are also related with the vortex shedding and scatting near the trailing edge. However, the difference with the low Reynolds number cases is that the formation of the vortices is governed by a feedback mechanism^{[3-4, 26]}. In Kingan and Pearse's^[26] feedback loop model, T-S waves propagate downstream at a phase velocity c_r, and generate large-scale vortices at the same frequencies. The large-scale vortices pass over the trailing edge and emit acoustic waves with a phase shifting of π. The emitted acoustic waves propagate upstream at velocity of (a_∞-U_∞) and excite the boundary layer. New T-S waves are induced as long as the acoustic disturbances are strong enough and in range of the unstable natural frequencies of the boundary layer. Then the feedback loop is closed. A formula to model the feedback loop is given as

$ \begin{aligned} f_{m} L\left(\frac{1}{c_{r}}+\frac{1}{a_{\infty}-U_{\infty}}\right)+\frac{1}{2} &=n, \\ &(n=1, 2, 3, \ldots) \end{aligned} $

(3)

where f_m is the modeled tonal frequencies, L is the distance between the trailing edge and the location where T-S waves first emerge, and n is an arbitrary integer. The phase velocity c_r is estimated using the real part of α in Eq.2, following the approach of Kingan and Pearse^[26]. Comparison of the total frequencies with the model predictions using Eq.3 are shown in Figure 11. Good agreements are obtained, indicating that the generation of the tonal noise are governed by the feedback loop mechanism. Feedback loops actually exist on both sides of the airfoil rather than just on the pressure side or on the suction side, which is different from Nash, Lowson, and Mcalpine^[6], Chong and Joseph^[12], Ikeda, Atobe, and Takagi^[13]'s investigation.

3 Conclusions

Two-dimensional direct numerical simulations are performed to investigate the tonal noise generation in flow past a NACA0015 airfoil at a low (Re_c=1×10⁴) and a moderate (Re_c=5×10⁴) Reynolds number. The airfoil are operated at three angles of attack, i.e. α =0°, 2°, 4°. Key concluding remarks are highlighted as below.

(1) The acoustic waves in all cases are emitted from the trailing edge of the airfoil, and display a type of bipolar nature. For the low Reynolds number cases the acoustic spectra is characterized by a single tone, whereas multiple tones are observed for the moderate Reynolds number cases. Due to the viscous effect, the large-scale vortices in the moderate Reynolds number cases are formed closer to the leading edge of theairfoil and exhibit much more unstable features than those in the low Reynolds number cases, especially on the pressure side of the airfoil.

(2) The generation of the acoustic waves is studied by using phase-averaging analysis and power spectral density of vorticity fluctuations. Results show that the passage of the large-scale vortices over the trailing edge, accompanying with unsteady vorticity variations, results in periodic pressure oscillations nearby which are observed as discrete tonal noise in the far field. It confirms that for the moderate Reynolds number cases, the vortex shedding on the suction and pressure side of the airfoil contributes separately to the generation of the acoustic waves, corresponding to an individual hump with different discrete frequencies in the acoustic spectra.

(3) To further explain the formation of the vortices, a linear stability analysis is performed. It is found that in all cases the vortex shedding frequency on each side of the airfoil is close to the most unstable frequency of the Tollmien-Schlichting (T-S) waves, indicating that the formation of the vortices is related to the growth of the unstable T-S waves. The single tone observed in the low Reynolds number cases is probably associated with the inherent hydrodynamic instability of the boundary layer, whereas the multiple tones in the moderate Reynolds number cases are considered to be associated with a feedback loop between the formation of the vortices and the excitation of the acoustic disturbances. With a comparison with Kingan and Pearse's^[26] feedback loop model, it shows that the feedback loops actually exist on both sides of the airfoil rather than just on the pressure side or on the suction side.

Acknowledgements: The authors acknowledge the funding support of National Basic Research Program of China (973 program) granted (2014CB744802, 2014CB744804) and the support of National Natural Science Foundation of China (11772194). The large-scale computations are supported by Center for High Performance Computing, Shanghai Jiao Tong University.