# Numerical Investigation of Tonal Trailing-Edge Noise Radiated by Low Reynolds Number Airfoils

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## Abstract

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## 1. Introduction

_{c}) on the airfoil transition from the tone-producing to no-tone-producing regimes, with the focus on the linear-stability analysis of the behavior of the laminar separation bubble (LSB) and evolution of viscous/inviscid instabilities during the transition process. The frequency selection process and the resulting spectral ladder-type frequency staging are examined as well. The final discussion of the obtained results is presented in Section 5.

## 2. Experimental and Numerical Approaches

#### 2.1. Experimental Approach

#### 2.2. Numerical Approach

#### 2.2.1. High-Fidelity ILES Model

_{∞}(γ is the specific heat ratio). The other variables in Equation (1) include the inviscid flux vectors defined by

#### 2.2.2. Numerical Implementation

^{−5}and Δx/c = 0.5 × 10

^{−3}, where c is the airfoil chord length. In terms of the wall units, y

_{w}

^{+}/c = 3.13 × 10

^{−5}is estimated for the characteristic flow condition, with M

_{∞}= 0.0465 and Re

_{c}= 1.4 × 10

^{5}. Such a grid refinement corresponds to the non-dimensional values of Δy

^{+}≈ 1 and Δx

^{+}= 20, with 12 grid points clustered in the region 0 < y

^{+}< 10. For 3D simulations, such grid parameters correspond to a high-resolution LES according to estimates in Wagner et al. [17] (p. 209). Such a grid is also finer compared to the mesh employed in the DNS study by Desquesnes et al. [5] conducted using a mesh with Δy/c = 3.8 × 10

^{−4}and Δx/c = 6 × 10

^{−3}. Note that the use of 2D analysis (much less computationally intensive, which is particularly critical for this parametric study) could be justified based on the assumption that, though inherently unsteady, the investigated flow regimes remain primarily laminar (with possible separation zones) and exhibit transitional features. Nevertheless, the next section will present a thorough comparison of the two approaches for the selected cases.

^{−5}corresponding to a physical time step of 0.675 × 10

^{−6}s, whereas the 3D simulations require a time step that is 4x finer, a 0.0225 × 10

^{−5}non-dimensionalized time or 0.16875 × 10

^{−6}s, to accommodate the fine grid spacing along the span. In all simulations, the solutions were first marched forward in time for 20 characteristics cycles to ensure transient processes have dissipated and a quasi-steady state is reached. The pressure signals were then recorded for over 65 cycles (720,000 iterations); hence, for the baseline setup, we collected the data sample for 0.487 s with a sampling rate of 33.6 kHz, achieving a frequency resolution of ∆f = 2.05 Hz.

## 3. Flow Configuration and Comparison of 2D vs. 3D Numerical Approaches

#### 3.1. Test Cases for NACA-0012 Airfoil

_{∞}= 0.0465) and Re

_{c}= 140,000 to closely mimic the experimental setup. The corresponding time-averaged U-velocity contours comparing the 2D and 3D ILES results are shown in Figure 2. Overall, the time-averaged U-velocity contours shown in Figure 2a,b are identical from the leading-edge (LE) to mid-chord due to the inherently 2D, and thus laminar, flow, as well as the absence of separation. Beyond the mid-chord, the flow similarities begin to diverge and the differences between the 2D and 3D ILES flow-fields become more apparent. Upon closer inspection, the 3D simulations reveal a noticeably smaller LSB and wake structures, which is a result of vortex breakdown captured in 3D ILES. To supplement the time-averaged U-velocity contours and characterize the thinner wake profile of the 3D simulations, near-wake U-velocity profiles are shown in Figure 3a. The results, which are measured 1% chord away from TE, confirms the size of the velocity deficit to be smaller in 3D simulations due to the energy redistribution along the span-wise direction.

#### 3.2. Spectral Analysis of Surface and Far-Field Pressure Data Samples

^{14}pressure samples over a time signal of 0.487 s, whereas experimental measurements were sampled over several seconds. Figure 5 compares the computed and measured spectra for various monitor points along the NACA-0012 surface (Figure 1) as well as the far-field, which is located 12.5 chords directly above the TE. Along the airfoil surface (near-field), the results show a strong correlation between the numerical predictions and measurements in predicting the main tonal signature, which occurred at 610 Hz. However, the overall results reveal that the 3D ILES provides a stronger correlation with the experiment in capturing the harmonics of the main tones, as shown in the experiment, as well as accurately predicting the peak frequency near the trailing edge at 0.945c. It should be noted that the overall broadband level is slightly reduced in the 3D ILES simulations, further improving the comparison with the experiments.

## 4. Results of the Parametric AFL Sensitivity Study

#### 4.1. Linear Stability Analysis

_{r}is the real spatial wavenumber and −α

_{i}determines the instantaneous growth rate at a given streamwise location. Once the growth rates are acquired at each location, they are integrated along the instability’s convection path to obtain the total growth factor N for a disturbance at a given frequency defined by

_{0}is the initial disturbance amplitude, and A is the amplitude of the disturbance at a given streamwise location. Similarly, x

_{0}and x denote the initial disturbance location and the final location, respectively.

_{ij}are the Jacobians of the flux vectors and R

_{0}is the Reynolds number used to normalize the equations (detailed in [23]). Note that, unlike the Orr–Sommerfeld equation, this formulation accounts for the flow-divergence effects in predicting the instability mode evolution. Substituting (9), the following set of equations are solved for the instability modal growth rates, −α

_{i}, with the no-slip wall boundary conditions applied at the airfoil surface and Dirichlet boundary condition applied in the free stream,

_{RMS}, was determined from a high-fidelity numerical analysis based on a sample of M selected tonal frequency modes,

_{RMS}curves may be easily related to the weighted modal growth rates,

#### 4.2. Variation in Angle of Attack for Re_{c} = 180,000: Tone-Producing Regimes

_{c}= 100,000 at 0° to 5° and found that the acoustic tone was identical to the vortex shedding frequency but was of lower frequency than the most amplified frequency predicted by LST. It should be noted that the tonal amplitudes are significantly higher on the suction surface, thus indicating instabilities on this side are the dominant source of pressure fluctuation growth shown earlier in Figure 8 and Figure 9.

_{c}laminar flow around a symmetric NACA-0012 airfoil, Tam and Ju [9] identified the shedding frequency with that of the most amplified near-wake K-H instability in the free shear layer. The TE scattering of such near-wake instability was claimed as the source of the shedding tone in the absence of AFL. Thus, such a wake instability mechanism may superimpose on the AFL mechanism of airfoil tonal noise production, with both related to the TE scattering of the free shear-layer K-H instabilities. This would point to the mutual resonance phenomenon where both mechanisms overlap and appear to enhance each other. When the flow past the airfoil is laminar with no AFL present, the remaining shedding tone appears at significantly lower amplitude, as demonstrated by Tam and Ju [9]. On the other hand, when the AFL mechanism is suppressed due to BL tripping or a low-intensity upstream turbulence, the observed acoustic spectrum still reveals a broadband hump centered around the shedding frequency (Golubev et al. [8]). It appears that the shedding mechanism is dominant and amplified with AFL presence due to the mutual interference and resonant interaction of the AFL-selected frequencies with shedding tone.

#### 4.3. Variation in Angle of Attack for Re_{c} = 180,000: No Tone-Producing Regimes

_{f}.

_{c}= 180,000, to elucidate the primary mechanism responsible for tonal noise suppression at an AoA greater than 6°. At α = 0°, shown in Figure 16, the flow remains in the transitional regime and its behavior is similar to that proposed by Brooks et al. [29]. Upon closer inspection of the Z-vorticity contours, the BL dynamics clearly reveals that the unsteady oscillatory motion of the vortex street at the TE is responsible for generating pressure pulses and the subsequent propagation of acoustic waves into the far-field. In addition to the vortex shedding noise, the interaction of the airfoil surface instabilities scattering into the wake is responsible for the additional noise contributions (neighboring peaks) shown in the surface spectra. The accompanying dilatation field shown in Figure 16 further confirms the acoustic wave to be emitted from the TE of the airfoil.

#### 4.4. Variation in Reynolds Number at α = 2°: Tone-Producing Regimes

_{c}at a fixed AoA of 2°. Clearly, all investigated cases are within the tone-producing region, and the cases of Re

_{c}= 144,000 (20 m/s) and Re

_{c}= 288,000 (40 m/s) selected for illustration exhibit well-defined tonal spectra in the near-field at 2 chords directly above the TE (Figure 19). BL statistical analysis reveals a direct correlation between increasing Re

_{c}and the decreasing chordwise extent of the suction side LSB (Figure 20 and Table 5), while the LSB on the pressure surface retains nearly the same size at the TE (with C

_{f}becoming more negative towards the TE with increasing velocity).

_{c}leads to a surge in amplification of the peak instability mode on the airfoil pressure side (Figure 21). At the same time, as the suction-side LSB shrinks while the instability saturation location shifts further upstream from the TE, the contribution of the suction surface to the radiated sound diminishes relative to that from the pressure side. It is expected that a further increase in flow velocity will result in the eventual suppression of the separation region from the suction side. When this occurs, the pressure-side LSB will provide the sole contributor to the radiated tonal noise. These trends are in agreement with the experimental findings of Golubev et al. [8], Yakhina et al. [10], and Pröbsting et al. [30], where tripping the BL on either side of the airfoil revealed a dominant tonal noise contribution from the suction side at the lower flow velocities, and from the pressure side at the higher flow velocities.

_{c}(<144,000) would lead to the formation of a fully laminar BL and result in the absence of the LSB. Consequently, without the LSB, the switch from T-S to K-H waves would not be possible and thus the rapidly amplifying instability modes needed for the self-sustained acoustic feedback can never be established. While it is possible for the shedding tone to still exist, it would likely be much lower in amplitude due to the lack of reinforcement from the feedback loop [9]. On the other hand, increasing the Re

_{c}(>288,000) would suggest transitioning to the fully turbulent BL flow, with both the acoustic feedback and the shedding tone fully suppressed.

#### 4.5. Acoustic Radiation Frequency Structure

_{c}= 128,000) uniform upstream flow, revealing the expected vortex roll up and shedding pattern. As described by Arbey and Bataille [3], the onset of instabilities occur at 0.86c due to the T-S waves that grow and transition to discrete vortices, which convect along the airfoil surface towards the TE. Once the vortices reach the TE, their interaction with the natural shedding of the wake coalesce to amplify the noise generated by the airfoil. As the mean flow velocity is increased to 19 m/s (Re

_{c}= 152,000) (Figure 22b), the overall behavior of the flow remains unchanged with the exception of the triggering of T-S to K-H instabilities. Unlike the 16 m/s (Re

_{c}= 128,000) case, at 19 m/s (Re

_{c}= 152,000), the transition from T-S to K-H instabilities appears further upstream as the increase in the flow velocity shifts the location of the separation region towards the LE. A similar pattern is exhibited as the mean flow velocity is increased to 21 m/s (Re

_{c}= 168,000) (Figure 22c) and 25 m/s (Re

_{c}= 200,000) (Figure 22d). It is worth mentioning that for all the flow regimes shown in Figure 22, the LSB is present and is the primary source to trigger instability amplification and the sustainment of the AFL.

_{c}on the tonal noise mechanism, spectral analysis is employed for various experimental probe locations along the airfoil surface (shown in Figure 1) to obtain the peak and neighboring tones. Details of the Fourier analysis performed to obtain the spectra are similar to those employed in the previous sections. However, for all subsequent near-field (2c above TE) spectra in this chapter, the results were obtained by sampling a signal that was 5 times longer, thus allowing for averaging of the FFT (using 5 segments) to smooth out the noise and accentuate the peaks of the signal. Figure 23 reveals the peak tonal noise (corresponding to Strouhal number St ~0.2) predicted by the numerical simulations for the mean flow velocity of 16 m/s (Re

_{c}= 128,000) to occur at a frequency of 636 Hz for every selected point along the airfoil, which correlates well with Paterson’s [1] empirical formula (14) for vortex shedding frequency,

_{c}= 152,000) (Figure 24), the peak tone shifts rightward in the spectrum and occurs at a higher frequency of 746 Hz (while maintaining St ~0.2). Again, correlating this value with Paterson’s formula (14) provides a good match and reveals that the increase from 16 m/s (Re

_{c}= 128,000) to 19 m/s (Re

_{c}= 152,000) would result in a shift from 574 Hz to 729 Hz. This confirms that the peak tones do in fact scale with ~U

^{1.5}, thus corresponding to the airfoil’s natural shedding frequency. Additionally, the increase in the flow velocity leads to neighboring tones with higher amplitudes that are more prominent and easily distinguished in the spectrum. Further increase in mean flow velocity to 21 m/s (Re

_{c}= 168,000) and 25 m/s (Re

_{c}= 200,000) shows a similar trend. Figure 25 and Figure 26 demonstrate the shift of the primary tones to higher frequencies, in agreement with Paterson’s formula (14).

_{c}= 200,000) (Figure 26) reveals that while the highest amplitude tone is observed at point A at 1199 Hz, correlating well with shedding frequency (14), and the probes at points B, C, and D show a peak tone at 1281 Hz. As suggested based on the LST analysis, this could be linked to the amplification of various instability modes occurring along the airfoil surface. Perhaps the peak shedding tone of 1199 Hz observed at the TE reduces in amplitude as it propagates upstream and interacts with probes B, C, and D. At the same time, it could be that the secondary peak of 1281 Hz results from a mode that is more amplified in the regions close to points B, C, and D but appears weaker at the TE, where the interaction of the saturated instability modes with the primary shedding mechanism dominates.

_{c}= 128,000) to 25 m/s (Re

_{c}= 200,000) directly affects the amplitudes of the neighboring tones surrounding the peak, and as a result, leads to a very pronounced peak. This comparison is clearly illustrated in Figure 24 and Figure 26 in which the former shows a very low amplitude with nearly suppressed neighboring tones, while the latter shows clearly defined tones. This observation correlates well with the Z-vorticity contours in Figure 22 by visually inspecting the vortices scattering in the wake and correlating the amplification of the neighboring tones with the amplitudes of the vortical structures developing along the airfoil surface. It is evident that at 16 m/s (Re

_{c}= 128,000) the vorticity contours are not as saturated compared to 25 m/s (Re

_{c}= 200,000), which is indicative of stronger vortex cores. These stronger vortex cores will, in turn, produce stronger acoustic waves and thus show higher amplitudes of the predicted spectral peaks.

_{s}is graphically determined by selecting a single highest amplitude peak tone recorded at the four monitor points along the airfoil (typically at the TE). The remaining neighboring tones, f

_{n}, are obtained by selecting the equally spaced tones surrounding the peak tone. The reconstructed ladder structure demonstrates that the solid line, f

_{s}, follows Paterson’s proposed U

^{1.5}and f

_{n}, dashed lines, follows the U

^{0.8}scaling. The agreement between Paterson’s formula (14) and the ILES results correlate very well and verifies the accuracy of the current numerical results as well as the existence of the AFL.

_{c}) along the airfoil surface, which can be used in the feedback-loop formula of Arbey and Bataille [3],

^{14}pressure slices extracted from the airfoil surface near the TE (0.85c–0.95c) over a period of 0.487 s to show a 2D spectrum representing the normalized frequency fδ/U

_{∞}vs. the normalized wavenumber, k

_{x}δ/2π, where δ is the BL thickness. The selected flow region, measuring 0.1c × 0.03c, was chosen to be near the TE based on the dilatation field revealing the TE to be the primary source of the airfoil acoustic radiation. Table 7 compares the computed (from numerical analysis) vs. predicted (Equation (15)) tonal frequencies for the four cases from 16 m/s to 25 m/s, to show a good agreement for the selected modal numbers. The remaining discrepancies are attributed to the change in the convective velocity along the airfoil surface not accounted in the employed Equation (15).

## 5. Discussion of Results

_{c}and correlated with position and size of the LSB. At low AoAs (α ≤ 6°), distinct tones were shown to be present that dominated the acoustics of the airfoil. However, the tones disappeared at higher AoAs (α > 6°) due to the LSB migration towards the LE, which allowed sufficient time for the flow to transition to fully turbulent regime with subdued AFL interactions. On the other hand, for a fixed AoA, the increase in flow Re

_{c}was associated with LSB shrinking on the suction side and an increasingly dominant contribution of the AFL on the pressure side to the tonal noise production. Additionally, a parametric investigation of the effect of Re

_{c}allowed recreating Paterson’s dual ladder-type frequency structure. Overall, the results of the parametric studies involving high-fidelity numerical studies and linear stability analysis clearly demonstrate that the rapid growth of the BL instability modes is associated with the presence of separation regions (i.e., LSB) on either side of the airfoil. The switch from the slowly growing T-S modes (associated with the BL viscous effects) to the fast-growing K-H modes (associated with the velocity gradients in the LSB detached shear layer) thus appears as a necessary condition for the strong multiple-tone-producing flow-acoustic interactions. This allows for an important modification to the AFL scenario originally proposed by Longhouse [33], as illustrated in Figure 29. The additionally required condition is the ability of the vorticity modes to sustain their presence and sufficient amplitude to the TE for the effective acoustic scattering process to take place. Hence, the LSB must be located close enough to the airfoil TE to prevent the subsequent destruction of the saturated coherent modes (rollers) through turbulent BL transition and spanwise mixing process.

_{c}laminar flow around a symmetric NACA-0012 airfoil, Tam and Ju [9] identified the shedding frequency with that of the most amplified near-wake K-H instability in the free shear layer. The TE scattering of such near-wake instability was claimed as the source of the shedding tone in the absence of AFL. It is plausible that such a wake instability mechanism may superimpose on the AFL mechanism of airfoil tonal noise production, with both related to the TE scattering of the nonlinearly saturated K-H modes. This points to the mutual resonant interaction wherein both mechanisms overlap and appear to enhance each other. To this end, when the airfoil flow is laminar with no AFL present, the remaining shedding tone appears at a significantly lower amplitude, as demonstrated by Tam and Ju [9]. On the other hand, when the AFL mechanism is suppressed due to BL tripping or a low-intensity upstream turbulence, the observed acoustic spectrum still reveals a broadband hump centered around the shedding frequency. It thus appears that the naturally primary shedding mechanism is dominant and amplified by the AFL presence due to the mutual interference and resonant interaction of the AFL-selected frequencies with the shedding tone. The latter essentially elevates that part of the spectrum of the amplified BL instability modes that appears the closest to the shedding tone. This may explain the divergence between the frequencies of the most amplified instability modes and the acoustically radiated peak tone coinciding with the shedding frequency. It is the interaction of the two mechanisms that produce the dual ladder-type acoustic tonal structure of Paterson et al. [1], with multiple AFL-selected tones staged around the shedding tone.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Locations of the Remote Microphone Probes (RMPs) on the NACA-0012 mock-up (red symbols) and the reference points for the analysis of wall-pressure fluctuations in the experiment (Golubev et al. [8]).

**Figure 2.**Time-averaged U-velocity contours: (

**a**) 2D; (

**b**) 3D Implicit Large-Eddy Simulations (ILES). Baseline case for NACA-0012 at a 0° angle of attack (AoA) and U = 16 m/s.

**Figure 3.**(

**a**) Near-wake U-velocity profile and (

**b**) comparison of predicted vs. measured surface pressure coefficient. Baseline case for NACA-0012 at a 0° AoA and U = 16 m/s.

**Figure 4.**Comparison of 2D (blue) and 3D ILES (magenta) vs. measured (red) suction-side boundary layer profiles. Averaged tangential U-velocity, NACA-0012 at a 0° AoA and U = 16 m/s.

**Figure 5.**Comparison of 2D (blue) and 3D ILES (magenta) vs. measured (red) airfoil surface and far-field pressure spectra. NACA-0012 U = 16 m/s. R/c represents the distance from the airfoil centerline and θ represents the angle measured from the chordline (counter-clockwise).

**Figure 6.**Time-averaged pressure contours from 2D NACA-0012 simulations (

**left**); corresponding separation regions (

**right**). U = 25 m/s.

**Figure 8.**Instability chordwise amplification at peak tonal frequency vs. statistical parameters for α = 0°.

**Figure 9.**Instability chordwise amplification at peak tonal frequency vs. statistical parameters for α = 6°.

**Figure 10.**Instability mode amplifications (blue line), location of the modal peaks (green line), and the peak acoustic frequency (dashed black line) for α = 0°, 2°, 4°, and 6°.

**Figure 14.**Instability chordwise amplification at various frequencies vs. statistical parameters for α = 12°.

**Figure 16.**NACA-0012 at an α = 0° AoA and Re

_{c}= 180,000. (

**Left**) Instantaneous Z-vorticity contours; (

**Right**) the dilatation field. Red circle indicates the origin of the acoustic wave to be at the trailing-edge (TE).

**Figure 17.**NACA-0012 at an α = 4° AoA and Re

_{c}= 180,000. (

**Left**) Instantaneous Z-vorticity contours; (

**Right**) the dilatation field. Red circle indicates the origin of the acoustic wave to be at the TE.

**Figure 18.**NACA-0012 at an α = 8° AoA and Re

_{c}= 180,000. (

**Left**) Instantaneous Z-vorticity contours; (

**Right**) the dilatation field. Red circle indicates the origin of the acoustic wave to be at the TE.

**Figure 21.**Instability chordwise amplification for Re

_{c}= 144,000 (

**left**) and Re

_{c}= 288,000 (

**right**).

**Figure 22.**Instantaneous Z-Vorticity contour: (

**a**) U = 16 m/s; (

**b**) U = 19 m/s; (

**c**) U = 21 m/s; (

**d**) U = 25 m/s.

**Figure 28.**Dispersion analysis (Wavefront wavenumber-frequency spectra) for a NACA-0012 at mean flow velocities of 16–25 m/s.

**Figure 29.**Suggested acoustic feedback loop (revised from Longhouse [32]).

Cases | Dimension | Δy/c | Δx/c | Δz/c |
---|---|---|---|---|

2D FINE | 1281 × 789 × 3 | 2.5 × 10^{−5} | 0.5 × 10^{−3} | - |

3D FINE | 1281 × 789 × 101 | 2.5 × 10^{−5} | 0.5 × 10^{−3} | 9.9 × 10^{−4} |

Airfoil | AoA (degree) | Velocity (m/s) | Re_{c} |
---|---|---|---|

NACA-0012 | 0, 2, 4, 6, 8, 10, 12 | 25 | 180,000 |

AoA | Suction Side | Pressure Side |
---|---|---|

0 | 70%–96% | 70%–96% |

2 | 53%–73% | 82%–100% |

4 | 35%–48% | No Separation |

6 | 9%–25% | No Separation |

Angle | Suction Side | Pressure Side |
---|---|---|

8 | 3%–12% | - |

10 | 2%–10% | - |

12 | 2%–8% | - |

Re_{c} | Suction Side | Pressure Side |
---|---|---|

144,000 | 52%c–80%c | 84%c–100%c |

180,000 | 53%c–74%c | 82%c–100%c |

216,000 | 54%c–69%c | 80%c–100%c |

252,000 | 55%c–66%c | 82%c–100%c |

288,000 | 57%c–64%c | 85%c–100%c |

U_{∞} (m/s), Re_{c} |
---|

16 (128,000), 19 (152,000), 21 (168,000), 25 (200,000) |

16 m/s (128,000) | 19 m/s (152,000) | |||||||

f, Hz computed (CFD) | 562 | 640 | 720 | f, Hz computed (CFD) | 633 | 746 | 835 | |

n = 10 | n = 12 | n = 13 | n = 10 | n = 11 | n = 12 | |||

f, Hz predicted, Equation (15) | 557 | 663 | 716 | f, Hz predicted, Equation (15) | 700 | 766 | 833 | |

21 m/s (168,000) | 25 m/s (200,000) | |||||||

f, Hz computed (CFD) | 699 | 820 | 939 | f, Hz computed (CFD) | 963 | 1099 | 1218 | |

n = 8 | n = 10 | n = 11 | n = 9 | n = 10 | n = 11 | |||

f, Hz predicted, Equation (15) | 696 | 860 | 942 | f, Hz predicted, Equation (15) | 992 | 1097 | 1202 |

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## Share and Cite

**MDPI and ACS Style**

Nguyen, L.; Golubev, V.; Mankbadi, R.; Yakhina, G.; Roger, M.
Numerical Investigation of Tonal Trailing-Edge Noise Radiated by Low Reynolds Number Airfoils. *Appl. Sci.* **2021**, *11*, 2257.
https://doi.org/10.3390/app11052257

**AMA Style**

Nguyen L, Golubev V, Mankbadi R, Yakhina G, Roger M.
Numerical Investigation of Tonal Trailing-Edge Noise Radiated by Low Reynolds Number Airfoils. *Applied Sciences*. 2021; 11(5):2257.
https://doi.org/10.3390/app11052257

**Chicago/Turabian Style**

Nguyen, Lap, Vladimir Golubev, Reda Mankbadi, Gyuzel Yakhina, and Michel Roger.
2021. "Numerical Investigation of Tonal Trailing-Edge Noise Radiated by Low Reynolds Number Airfoils" *Applied Sciences* 11, no. 5: 2257.
https://doi.org/10.3390/app11052257