# Influence of Swept Blades on Low-Order Acoustic Prediction for Axial Fans

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## Abstract

**:**

## 1. Introduction

## 2. Experimental Setup

## 3. Numerical Simulation and RANS Extraction Procedure

#### 3.1. Inlet Turbulence

#### 3.2. Boundary-Layer Parameters

## 4. Theoretical Background on Noise Prediction Methodology

#### 4.1. Noise Emitted by Rotating Blades

#### 4.2. Leading-Edge Noise Formulation

#### 4.3. Trailing-Edge Noise Formulation

#### 4.4. Generalized Corcos’ Model

#### 4.5. Semiempirical Wall-Pressure Models

#### 4.5.1. Goody’s Model

#### 4.5.2. Rozenberg’s Model

- $a=[2.82{\Delta}_{*}^{2}{\left(\right)}^{6.13}e]+1$,
- $b=2.0$,
- $c=0.75$,
- $d=4.76{\left(\right)}^{1.4}0.75$,
- $e=3.7+1.5{\beta}_{c}$,
- $f=8.8$,
- $g=-0.57$,
- $h=min\left(\right)open="("\; close=")">3,19/\sqrt{{R}_{T}}$,
- $i=4.76$.

#### 4.5.3. Lee’s Model

## 5. Acoustic Far-Field Results

#### 5.1. Noise Distribution over the Strips

#### 5.2. Leading-Edge Upstream Extraction Location

#### 5.3. LE and TE Noise Comparison with Experimental PSD

#### 5.4. Sensitivity Study

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Latin letters: | |

B | number of the blades |

b | half-chord aligned with x and ${U}_{0}$ |

${b}^{\prime}$ | rotated half-chord aligned with ${x}^{\prime}$ |

${b}_{c}$ | Corcos model constant |

c | chord aligned with ${U}_{0}$ |

${c}^{\prime}$ | rotated chord aligned with ${x}^{\prime}$ |

${c}_{0}$ | speed of sound |

${C}_{p}$ | pressure coefficient |

$E[-]$ | Fresnel Integral |

$\mathbf{K}$ | wavenumber vector |

${K}_{c}^{\prime}$ | convective wavenumber parallel to ${x}^{\prime}$ |

${K}_{x}^{\prime}$ | aerodynamic wavenumber parallel to ${x}^{\prime}$ |

${K}_{y}^{\prime}$ | aerodynamic wavenumber parallel to ${y}^{\prime}$ |

${k}_{a}$ | acoustic wavenumber |

${k}_{e}$ | average wavenumber of the energy-containing eddies |

${k}_{t}$ | turbulent kinetic energy |

${k}_{\omega}$ | convective wavenumber $\omega /{U}_{c}$ |

${\mathcal{L}}^{LE}$ | leading-edge aeroacoustic transfer function |

${\mathcal{L}}^{TE}$ | trailing-edge aeroacoustic transfer function |

L | span of a strip aligned with the radius |

${L}^{\prime}$ | rotated span of a strip aligned with ${y}^{\prime}$ |

${L}_{B}$ | radial blade length |

$LE$ | leading edge |

${l}_{y}$ | spanwise correlation length of wall-pressure fluctuations |

${M}_{i}$ | ${U}_{i}/{c}_{0}$, Mach number based on the i-th mean velocity component |

${M}_{r}$ | Mach number of the source relative to the fluid |

$m,n$ | Butterworth filter order coefficients |

${p}_{\infty}$ | laboratory pressure reference |

${R}_{T}$ | ratio of timescales of pressure |

r | fan radial distance |

R | radius of the fan |

${S}_{0}$ | listener’s corrected distance |

${S}_{pp}$ | far-field sound PSD of the fan |

${S}_{pp}^{\Psi}$ | single-strip airfoil noise |

${S}_{pp}^{LE}$ | single-strip leading-edge airfoil noise |

${S}_{pp}^{TE}$ | single-strip trailing-edge airfoil noise |

$TE$ | trailing edge |

$\overline{{u}^{2}}$ | mean square of the velocity fluctuations |

U | boundary-layer velocity |

${U}_{0}$ | local entrainment velocity |

${U}_{c}$ | convective velocity |

${U}_{e}$ | external boundary-layer velocity |

${U}_{max}$ | boundary-layer maximum velocity |

${U}_{x}$ | rotated velocity component parallel to ${x}^{\prime}$ |

${U}_{y}$ | rotated velocity component parallel to ${y}^{\prime}$ |

$\mathbf{x}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}(X,Y,Z)$ | listener’s position |

$(x,y,z)$ | local reference frame with x axis aligned with ${U}_{0}$ |

$({x}^{\prime},{y}^{\prime},z)$ | rotated local reference frame with ${y}^{\prime}$ parallel to the LE and TE edges |

${x}_{c}$ | strip-chord distance parallel to c |

$\mathbf{y}$ | vectorial location of the noise source |

${y}^{+}$ | boundary-layer dimensionless wall distance |

${y}_{c}$ | boundary-layer vertical distance |

Greek letters: | |

$\alpha $ | ${U}_{0}/{U}_{c}$ |

${\beta}_{0}$ | compressibility factor |

${\beta}_{i}$ | compressibility factor based on i-th mean velocity component |

$\gamma $ | Gamma function |

$\delta $ | boundary-layer thickness |

${\delta}^{*}$ | boundary-layer displacement thickness |

$\theta $ | boundary-layer momentum thickness |

$\Theta $ | directivity angle in the X-Z plane |

${\Lambda}_{f}$ | turbulent integral length scale |

${\Phi}_{ww}({k}_{x},{k}_{y})$ | two-wavenumber velocity fluctuations spectrum |

${\varphi}_{pp}(\omega )$ | single-point frequency spectrum of wall-pressure fluctuations |

$\Pi (\omega ,{k}_{x},{k}_{y})$ | two-wavenumber-frequency spectral density of wall-pressure fluctuations |

${\Pi}_{0}(\omega ,{k}_{y})$ | one-wavenumber-frequency spectral density of wall-pressure fluctuations |

$\Psi $ | fan azimuthal position |

$\psi $ | sweep angle defined between x and ${x}^{\prime}$ |

${\rho}_{\infty}$ | laboratory density reference |

${\tau}_{w}$ | wall-shear stress across the boundary layer |

$\omega $ | angular frequency |

${\omega}_{e}$ | source-emitted frequency |

${\omega}_{t}$ | turbulent kinetic energy specific dissipation rate |

$\Omega $ | fan velocity rotation |

Others: | |

$\overline{(\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}})}$ | normalization by the rotated half chord, ${b}^{\prime}$ |

$\widehat{(\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}})}$ | normalization by ${k}_{e}$ |

${(\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}})}^{\prime}$ | sweep-angle rotation |

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**Figure 1.**The relative velocity is plotted for an unsteady DES computation onto the isosurface obtained with the Q-criterion technique; large vortical structures are depicted in particular close to the ring and in the trailing-edge recirculating region near the hub.

**Figure 2.**(

**a**) ALCOVES anechoic chamber: upstream room with B&K microphones and open-rotor configuration from [18]. (

**b**) Suction-side 3D CAD model of the Valeo forward-skewed fan employed in the steady RANS simulation.

**Figure 3.**The RANS-based numerical results show similar azimuthal features over the 7 non-equally-distributed fan blades, making it possible to deal with them separately: in (

**a**), the relative velocity is illustrated on a plane normal to the fan rotating axis, showing higher velocities at the blade trailing-edge tips; in (

**b**), the distribution of pressure indicates that the most loaded zones are the blade leading-edge tips.

**Figure 4.**Inlet turbulence parameters are extracted from the steady-RANS computation in upstream of the blade leading-edge line and depicted as functions of the spanwise normalized distance y/L

_{B}: in (

**a**), the mean square of the velocity fluctuations $\overline{{u}^{2}}$ is depicted; in (

**b**), the characteristic length scale of the turbulent eddies ${\Lambda}_{f}$ is illustrated.

**Figure 5.**Three isoradial cuts are plotted at 3 spanwise locations y/L

_{B}: (

**a**) the boundary-layer velocity profiles are depicted as functions of the non-dimensional boundary-layer distance y

_{c}/c with the red dots corresponding to the boundary layer thickness δ; (

**b**) the negative pressure coefficient C

_{p}is plotted as function of the chordwise nondimensional coordinate x

_{c}/c.

**Figure 6.**(

**a**) From [15], the rotating local airfoil reference frame is denoted. The fixed reference frame $\mathrm{x}=(X,Y,Z)$ defines the observer’s position. (

**b**) Sweep-angle definition over the leading edge of the fan blade; the rotated local reference frame $({x}^{\prime},{y}^{\prime},z)$ is determined.

**Figure 7.**For the leading-edge case, the blade is divided into 10 segments: in (

**a**), a classical unswept leading-edge formulation can be used; in (

**b**), the effect of the sweep is evaluated with blade strips which have locally parallel leading-edges, following the blade forward-skewed curvature.

**Figure 8.**For the trailing-edge case, the blade is divided into 10 segments: in (

**a**), a classical unswept trailing-edge formulation can be used; in (

**b**), the effect of the sweep is evaluated with blade strips which have locally parallel trailing-edges, following the blade forward-skewed curvature.

**Figure 9.**Convergence study on the number of strips to use in order to acceptably discretize the noise sources on the blade span: (

**a**) leading-edge noise prediction case, (

**b**) trailing-edge noise prediction case.

**Figure 10.**Far-field sound power-spectral density (PSD) distribution over the blade span. The dashed curves represent the classical Amiet’s theory, whilst the solid ones take into account the varying sweep-angle: (

**a**) leading-edge noise prediction case, (

**b**) trailing-edge noise prediction case.

**Figure 11.**The turbulent kinetic energy content is shown in (

**a**); colored concentric circles represent the isoradial locations at which the blade is cut in order to study the overall SPL of the strips shown in (

**b**). Here, the overall SPL is depicted against the upstream chord distance from the leading-edge point.

**Figure 12.**PSD of the far-field emitted sound by the fan at its nominal working point: the solid red line representing the summation of trailing-edge (TE) and leading-edge (LE) noise contributions for the swept case (SW) is in fairly good agreement with the experimental curve in solid blue, especially at high frequency, resulting into a better description of the noise emission with respect to the unswept (UN) classical Amiet’s formulation shown in dashed line. In this plot, Lee’s model and Corcos’ model with $m=1$ and $n=1$ are employed for the TE noise prediction, whilst the von Kármán’s model is used for the LE noise prediction.

**Figure 13.**A parametric study is proposed for the SW case by comparing the predicted results with the measured ones: in (

**a**), 3 semiempirical wall-pressure models Lee’s, Rozenberg’s, and Goody’s are compared for the TE case; 2 turbulence velocity fluctuations models: von Kármán’s and Liepmann’s are compared for the LE case. In (

**b**), several combinations of the Corcos’ order coefficients m and n are tested out.

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## Share and Cite

**MDPI and ACS Style**

Zarri, A.; Christophe, J.; Moreau, S.; Schram, C.
Influence of Swept Blades on Low-Order Acoustic Prediction for Axial Fans. *Acoustics* **2020**, *2*, 812-832.
https://doi.org/10.3390/acoustics2040046

**AMA Style**

Zarri A, Christophe J, Moreau S, Schram C.
Influence of Swept Blades on Low-Order Acoustic Prediction for Axial Fans. *Acoustics*. 2020; 2(4):812-832.
https://doi.org/10.3390/acoustics2040046

**Chicago/Turabian Style**

Zarri, Alessandro, Julien Christophe, Stéphane Moreau, and Christophe Schram.
2020. "Influence of Swept Blades on Low-Order Acoustic Prediction for Axial Fans" *Acoustics* 2, no. 4: 812-832.
https://doi.org/10.3390/acoustics2040046