# Experimental and Analytical Investigation of the Tonal Trailing-Edge Noise Radiated by Low Reynolds Number Aerofoils

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## Abstract

**:**

## 1. Introduction

- the possible reason of discrepancies in experimental and numerical studies,
- the role of the separation bubble in tonal noise generation,
- the experimental setup issues,

## 2. Experimental Set-Up and Instrumentation

#### 2.1. Aerofoil Installation

#### 2.2. Aerofoil Instrumentation and Methodology

#### 2.2.1. Remote-Microphone Probes

#### 2.2.2. Cross-Spectral Analysis of Wall-Pressure Data

- -
- Two signals of pure acoustic nature for any set of two probes or microphones are perfectly correlated and as such lead to a coherence of about 1.
- -
- Two wall-pressure probes close to each other in the chordwise/streamwise direction usually provide significant coherence levels under most convected hydrodynamic excitations, as expected from the almost frozen character of boundary-layer turbulence over short distances. The coherence decreases with increasing probe separation. In contrast, the wall-pressure field of laminar instabilities is probably as coherent along the aerofoil chord as the one of acoustic waves.
- -
- The coherence between signals from wall-pressure probes aligned in the spanwise direction close to the trailing edge beneath a turbulent boundary layer gives access to the spanwise correlation length involved in most statistical trailing-edge noise models [38]. Its use in the present study will be discussed in Section 5.

#### 2.2.3. Time-Frequency Analysis

#### 2.2.4. Hot-Wire Anemometry and Flow Visualization

## 3. Results and Discussion

#### 3.1. Far-Field Measurements

#### 3.1.1. Radiation Maps

#### 3.1.2. Spectrograms

#### 3.2. Roles of Pressure-Side and Suction-Side Boundary Layers in the Tonal Noise Generation

#### 3.3. Characterization of Separation Bubbles

#### 3.4. Cross-Inspection of Velocity and Pressure Spectra

#### 3.5. Influence of the Upstream Turbulence

#### 3.6. Cross-Spectrum Analysis of the Wall-Pressure Measurements

#### 3.6.1. Chordwise Cross-Spectrum Analysis

#### 3.6.2. Spanwise Analysis of the Wall-Pressure Field

## 4. Comparison with Rotating-Blade Configurations

## 5. Analytical Prediction of the Tonal Noise Emission

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Tested aerofoil shapes in chordwise and normal coordinates $(x,y)$. Common chord length $c=12$ cm.

**Figure 3.**Far-field sound spectra measured normal to the flow direction in the mid-span plane with aerofoil alone (

**black**) and with aerofoil and splitter-plate (

**grey**). NACA-0012 aerofoil at zero angle of attack. (

**a**) 16 m/s, (

**b**) 25 m/s. Jet-splitter noise marked with the dashed arcs.

**Figure 4.**Pin-hole locations and probe labels of the wall-pressure Remote Microphone Probes (RMP) on the NACA-0012 mock-up. (

**a**) Right-bank side; (

**b**,

**c**) left-bank side. Relative position of the tripping device on the right-bank side indicated as the black strip in subplot (

**a**). Numbers of probes in red, spacing between probes in black.

**Figure 5.**Pin-hole locations and probe labels of the wall-pressure RMP on the SD7003 mock-up. (

**a**) Pressure side; (

**b**,

**c**) suction side. Numbers of probes in red, spacing between probes in black.

**Figure 6.**Frequency-flow speed maps of Laminar-Boundary-Layer (LBL)-wave radiation for the NACA-0012 aerofoil at zero angle of attack (

**a**) and for the SD7003 aerofoil at ${2}^{\circ}$ geometrical angle of attack (

**b**). Clean aerofoils. Dominant tone frequencies marked by red circles. 8 values of the flow speed in Table 1 featured by vertical segments. White-dotted lines generated by Equation (3).

**Figure 7.**Sound-pressure spectra extracted from the map in Figure 6b showing multiple tones for the SD7003 aerofoil. Values of the averaged frequency jumps for series of tones indicated on the plots (dashed vertical segments).

**Figure 8.**Time-frequency analysis, (

**left**) spectrograms, (

**right**) averaged far-field spectra; (

**a**,

**b**) for the SD7003 aerofoil and (

**c**,

**d**) for the NACA-0012 aerofoil at $\alpha ={0}^{\circ}$. Note that for case (

**a**), two averaged spectra are shown for the two states of the switching regime.

**Figure 9.**Frequency-versus-flow speed maps of LBL-wave radiation for the NACA-0012 aerofoil at $\alpha ={5}^{\circ}$. (

**a**) Tripping on the pressure side; (

**b**) tripping on the suction side; (

**c**) clean configuration. Dominant tone frequencies marked by red circles. Black and white arrows and segments pointing to the steps of the ladder-type structures in subplots (

**a**,

**b**) reproduced for comparison in (

**c**).

**Figure 10.**Flow visualization (

**a**) LE—leading edge, TE—trailing edge. Velocity profiles produced by the HWA (

**b**) for the suction side of the SD7003 aerofoil at $\alpha ={2}^{\circ}$ and 19 m/s for which tonal noise is observed. Upper plots: mean velocity. Lower plots: rms velocity.

**Figure 11.**Separated-flow areas on the NACA-0012 aerofoil at geometrical angles of attack $\alpha ={0}^{\circ}$ (

**a**), $\alpha ={5}^{\circ}$ (${\alpha}^{*}={1.75}^{\circ}$) (

**b**), and $\alpha ={10}^{\circ}$ (

**c**) for various flow speeds. Upper and lower sides as suction and pressure sides in subplots (

**b**) and (

**c**), respectively. HWA (dark bars) and oil-visualization (grey bars) results. Indicative simulation results at ${\alpha}^{*}={2}^{\circ}$ as black dotted lines [27].

**Figure 12.**Separated-flow areas on the suction side of the SD7003 aerofoil at geometrical angles of attack $\alpha =-{2}^{\circ},\phantom{\rule{4pt}{0ex}}{0}^{\circ},\phantom{\rule{4pt}{0ex}}{2}^{\circ}$, and ${5}^{\circ}$ at 19 m/s (

**a**), 25 m/s (

**b**), and 33 m/s (

**c**). HWA (

**dark bars**) and oil-visualization (

**grey bars**) results.

**Figure 13.**(

**a**) Geometrical angle of attack versus Reynolds number for the NACA-0012 aerofoil and the SD7003 aerofoil. Filled circles show the limits of the tonal-noise regime; empty circles point to the onset of low-amplitude noise. Green area is the primary emission for the NACA-0012 aerofoil, red and orange areas are tonal noise generated by pressure or suction sides of the NACA-0012, respectively, grey area is tonal noise generated by suction side of the SD7003 aerofoil. (

**b**) Effective angle of attack versus Reynolds number for the NACA-0012 aerofoil compared with previous investigations.

**Figure 14.**Compared velocity spectra from HWA (dB, ref.1 m/s), wall-pressure spectra from some RMPs and far-field acoustic spectrum (both dB, ref. $2\times {10}^{-5}$ Pa) for the SD7003 aerofoil. Geometrical angle of attack ${2}^{\circ}$, ${U}_{\infty}=19$ m/s. Pressure side (

**a**) and suction side (

**b**). Emphasis on the tone at 770 Hz.

**Figure 15.**Hot-wire velocity spectra (dB, ref.1 m/s) for the residual wind-tunnel turbulence and the grid-generated turbulence. Reference acoustic velocity levels indicated for sound pressure levels of 74 dB (0.1 Pa) and 94 dB (1 Pa). ${U}_{\infty}=25$ m/s. Model von Kármán spectrum with parameters indicated on the plot.

**Figure 16.**(

**a**) Frequency-flow speed chart for the NACA-0012 aerofoil at zero angle of attack with small-scale turbulence; (

**b**) compared far-field sound spectra of the clean flow (grey) and with upstream turbulence (black). Flow speeds 8 m/s and 25 m/s. Both pairs of spectra are shifted by 20 dB from each other.

**Figure 17.**Signal processing for pairs of chordwise RMPs. (

**a**) wall-pressure PSD in equivalent arbitrary decibels, first probe as grey dashed line, second one as black line. (

**b**) Coherence plots shifted by 1 for clarity between adjacent pairs. (

**c**) Phase of the cross-spectrum in linear frequency scale, with resolutions of 1 Hz (grey) and 8 Hz (black). Suction-side probes of the SD7003 aerofoil at $\alpha ={2}^{\circ}$ and ${U}_{\infty}$ = 25 m/s.

**Figure 18.**Signal processing for pairs of chordwise RMPs. (

**a**) Wall-pressure PSD in equivalent arbitrary decibels, first probe as grey dashed line, second one as black line. (

**b**) Coherence plots shifted by 1 for clarity between adjacent pairs. (

**c**) Phase of the cross-spectrum in linear frequency scale, with resolutions of 1 Hz (

**grey**) and 8 Hz (

**black**). NACA-0012 aerofoil at $\alpha ={0}^{\circ}$ and ${U}_{\infty}$ = 30 m/s.

**Figure 19.**Spanwise-coherence surface for LBL-wave radiation with (

**a**) and without (

**b**) acoustic feedback (fine-grid turbulence). NACA-0012 aerofoil at ${U}_{\infty}=16$ m/s and zero angle of attack. Right-bank side probes set B. Dashed line pointing to the dominant tone frequency. Frequency axis pointing from right to left for clarity.

**Figure 20.**Time signals from the spanwise set B of RMPs. NACA-0012 aerofoil at 16 m/s (

**a**) and 30 m/s (

**b**). Zero angle of attack.

**Figure 22.**Theoretical modulation spectra corresponding to Grosche and Stiewitt’s experiment [2]. Blade segments of radii 88% (grey bars) and 92% (empty bars) of the tip radius.

**Figure 23.**Evidence of non-tonal LBL-wave radiation from a small-scale low-speed fan at various rotational speeds. A-weighted sound levels. Scaling by the blade-passing frequency (

**a**) and by the power 1.5 of the rotational speed (

**b**).

**Figure 24.**Sample simulated growth of LBL instabilities on the NACA-0012 aerofoil with linear stability theory, from Nguyen et al. [27]. (

**a**) Amplitude of oscillations in arbitrary units, as a function of the chordwise coordinate in the aft part of the aerofoil for 1550 Hz, showing saturation. Trailing edge at x = 12 cm. (

**b**) Same data in logarithmic scale. Last RMP located at x = 10.5 cm (vertical segment). $\alpha ={0}^{\circ}$, ${U}_{\infty}$ = 25 m/s.

Nr. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|---|

3 Speed ${U}_{\infty}$ (m/s) | 8 | 11 | 16 | 19 | 21 | 25 | 30 | 33 |

Reynolds number $Re\phantom{\rule{0.166667em}{0ex}}(\times {10}^{5})$ | $0.63$ | $0.87$ | $1.26$ | $1.5$ | $1.6$ | 2 | $2.4$ | $2.6$ |

**Table 2.**Separation areas. Illustrative table for the NACA-0012 aerofoil. $\alpha $ geometrical angle of attack; ${U}_{\infty}$ flow velocity; ss and ps suction and pressure sides, respectively; FV flow visualization; HWA hot-wire anemometry. The first value of the percentage points to the beginning of the separation area and the second one to the end.

${\mathit{U}}_{\infty}$ | Side | 19 m/s | 25 m/s | 33 m/s | ||
---|---|---|---|---|---|---|

$\mathit{\alpha}$ | FV | HWA | FV | FV | ||

${0}^{\circ}$ | 54%, reversed | 42–92% | 54–85% | 57–80% | ||

58–100% | No tones | |||||

${5}^{\circ}$ | ss | 40–80% | 33–88% | 45–62% | no measurement | |

33–79% | ||||||

ps | 60–100% | 67–100% | 62%, reversed | no measurement | ||

${10}^{\circ}$ | ss | no measurement | no measurement | no measurement | 10–17% | |

ps | 80%, reversed |

**Table 3.**Separation areas. Illustrative table for the SD-7003 aerofoil. $\alpha $ geometrical angle of attack; ${U}_{\infty}$ flow velocity; ss and ps suction and pressure sides, respectively; FV flow visualization; HWA hot-wire anemometry. The first value of the percentage points to the beginning of the separation area and the second one to the end.

${\mathit{U}}_{\infty}$ | Side | 19 m/s | 25 m/s | 33 m/s | ||
---|---|---|---|---|---|---|

$\mathit{\alpha}$ | FV | HWA | FV | FV | ||

${0}^{\circ}$ | ss | 58–100% | 67–100% | 72–100% | ||

ps | attached | attached | attached | |||

${2}^{\circ}$ | ss | 54–100% | 46–92% | 60–90% | 54–80% | |

ps | attached | 33–96% | attached | attached | ||

${5}^{\circ}$, | ss | 42–62% | 33–75% | 46–62% | no measurement | |

No tones | ps | attached | 33–98% | attached | no measurement | |

$-{2}^{\circ}$ | ss | 71%, reversed | no measurement | 77%, reversed | 80%, reversed | |

ps | attached | no measurement | attached | attached |

**Table 4.**Tone levels as predicted with analytical modeling (AM) and measured in the far-field (FF). NACA-0012 aerofoil at 16 m/s and zero angle of attack. Probe numbers, related source levels (PSD) and convection speed are indicated.

Frequency 556 Hz | Frequency 635 Hz | ||||||
---|---|---|---|---|---|---|---|

PSD, dB | ${\mathit{U}}_{\mathit{c}}/{\mathit{U}}_{\infty}$ | AM, dB | FF, dB | PSD, dB | AM, dB | FF, dB | |

pr 4 | 94.4 | 0.29 | 47.2 | 49 | 107.2 | 60.3 | 61.5 |

0.5 | 49.6 | 62.7 | |||||

0.7 | 51 | 64.2 | |||||

pr 23 | 0.29 | 109.2 | 62.3 | ||||

0.5 | 64.7 | ||||||

0.7 | 66.2 |

**Table 5.**Tone levels as predicted with analytical modeling (AM) and measured in the far-field (FF). NACA-0012 aerofoil at 25 m/s and zero angle of attack. Probe numbers, related source levels (PSD), and convection speed are indicated.

Frequency 1200 Hz | Frequency 1333 Hz | ||||||
---|---|---|---|---|---|---|---|

PSD, dB | ${\mathit{U}}_{\mathit{c}}/{\mathit{U}}_{\infty}$ | AM, dB | FF, dB | PSD, dB | AM, dB | FF, dB | |

pr 4 | 99.6 | 0.29 | 55.4 | 49.9 | 109.5 | 65.1 | 56.7 |

0.5 | 57.8 | 67.5 | |||||

0.7 | 59.3 | 69 | |||||

pr 23 | 0.29 | 107 | 62.6 | ||||

0.5 | 65 | ||||||

0.7 | 66.5 |

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## Share and Cite

**MDPI and ACS Style**

Yakhina, G.; Roger, M.; Moreau, S.; Nguyen, L.; Golubev, V.
Experimental and Analytical Investigation of the Tonal Trailing-Edge Noise Radiated by Low Reynolds Number Aerofoils. *Acoustics* **2020**, *2*, 293-329.
https://doi.org/10.3390/acoustics2020018

**AMA Style**

Yakhina G, Roger M, Moreau S, Nguyen L, Golubev V.
Experimental and Analytical Investigation of the Tonal Trailing-Edge Noise Radiated by Low Reynolds Number Aerofoils. *Acoustics*. 2020; 2(2):293-329.
https://doi.org/10.3390/acoustics2020018

**Chicago/Turabian Style**

Yakhina, Gyuzel, Michel Roger, Stéphane Moreau, Lap Nguyen, and Vladimir Golubev.
2020. "Experimental and Analytical Investigation of the Tonal Trailing-Edge Noise Radiated by Low Reynolds Number Aerofoils" *Acoustics* 2, no. 2: 293-329.
https://doi.org/10.3390/acoustics2020018