#
Preliminary Design Guidelines for Considering the Vibration and Noise of Low-Speed Axial Fans Due to Profile Vortex Shedding^{ †}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

_{c}= 5 × 10

^{4}is assumed herein, while the upper limit is determined by the critical Reynolds number of the natural laminar-to-turbulent transition. When PVS is discussed for low-speed fans, as in the present paper, incompressible flow is considered by implying a Mach number of ≤0.3.

_{0}and the TE thickness d

_{TE}:

_{c}= 9 × 10

^{4}, 1.3 × 10

^{5}, 2.6 × 10

^{5}). All of the aforementioned observations—both the isolated blade profile and rotor consideration—suggest that the following blade features tend to increase the inclination for the occurrence of well-detectable tonal PVS: high aspect ratio (AR), low solidity, moderate twist, and constant blade chord. These parameters are typical for propeller-type fans [21]. Such propeller-type fans, where low-solidity is characteristic over a significant portion of the span, have been designed, for instance, by [22,23].

## 2. Blade Vibration: An Overview

_{b}is the density of the blade material. The method of variable separation can be used to produce the free vibration solution. By utilizing the proper initial and boundary conditions, the i-th eigenfrequency (see later) and normal mode shape [31] can be expressed as follows:

_{i}l ≈ (2i − 1)π/2. For illustrative examples, Figure 2 qualitatively presents the shapes for some bending modes, generated on the basis of Equation (4). The vertical axis represents the normal mode (i.e., dimensionless displacement) and the horizontal axis shows the dimensionless length of the beam.

_{t}is the torsional stiffness or torsional constant; and I

_{p}is the polar moment of area of the blade section.

#### 2.1. Analytical Treatment

#### 2.1.1. Bending Modes

_{B,i}constant into Equation (6). The values of K

_{B,i}for the first three bending modes are K

_{B,1}= 0.560, K

_{B,2}= 3.506 and K

_{B,3}= 9.819, respectively.

#### 2.1.2. Torsional Modes

_{t}—by utilizing the material and geometrical parameters of the blade. To calculate the I

_{t}torsional constant for a flat plate, the following relation is given in [26]:

_{t}is a mechanical constant which can be obtained from Figure 4.2 of [26] or calculated applying the formula—being in accordance with the former literature—e.g., [36]. However, the aforementioned alternatives for determining the torsion constant are related not to a cambered but to a flat plate, inhibiting their direct application in our case study. To overcome this problem, based on [36,37,38], the torsional constant of a thin-walled open tube cross-section of uniform thickness can be expressed as:

#### 2.2. Finite Element Method (FEM)

#### 2.2.1. Geometry, Materials, and Elements

^{3}, a Young’s modulus of 200 GPa, and a Poisson’s ratio (ν

_{P}) of 0.3.

#### 2.2.2. Mesh Convergence

#### 2.3. Comparison of the Results

_{analytical}in comparison to f

_{FEM}.

## 3. Blade Mechanics: An Exemplary Case Study

#### 3.1. The Importance of the First-Order Bending Mode

#### 3.2. Eigenfrequency

- (a)
- At moderate Reynolds numbers and angles of attack (α), the cambered plate produces reasonably high C
_{L}, that is comparable with an airfoil profile, i.e., RAF-6E [43], thus enabling the design of blades of relatively high specific performance, i.e., utilizing the loading capability of the blade sections. - (b)
- At 8% relative curvature, the lift-to-drag (LDR) is near the maximum among the cambered plates of various relative camber, thus enabling the design for reasonably high efficiency [44].
- (c)
- In accordance with the aforementioned practical aspects, it is a cambered plate of 8% relative camber for which hot-wire measurement data are made available by the present authors on PVS at different free-stream velocities and angles of attack [13]. As per the illustration, Figure 4 shows C
_{L}, C_{D}, and LDR values as a function of the angle of attack for the 8% cambered plate.

_{b}, is termed herein the wave propagation speed, and denoted as a

_{b}. This is actually the acoustic wave propagation speed in a long one-dimensional fictitious beam made of the blade material. The resultant swinging pattern is shown in Figure 6.

#### 3.3. Second Moment of Area

_{cambered}was obtained for the cambered plate-section using an analytical integration process known from basic solid-state mechanics [45]. As background information for the reader, the values of K

_{1}are presented in Figure 7 for representative relative thickness and relative camber (h/c) values. For the fitted curves in Figure 7, K

_{1}was calculated for fixed t/c values for uniform steps of 0.01 h/c over the entire h/c range. In the blade design presented later, K

_{1}(t/c = 0.02; h/c = 0.08) = 18.07 was used, in accordance with the previously selected blade geometrical parameters.

_{b}= K

_{b}(t/c = 0.02; h/c = 0.08) = 1.23 is the blade mechanics coefficient. Case studies considering other t/c and h/c values can be carried out using Figure 7.

## 4. PVS-Affected Rotor: An Exemplary Case Study

- (a)
- The rotor blading tends to exhibit a PVS of spanwise constant frequency, and thus, is theoretically presumed to realize large-scale, spatially coherent vortices over the dominant portion of the blade span. Such coherent vortices are assumed to cause spatially correlated, narrowband noise, as well as mechanical excitation over the dominant part of span at a given frequency. The condition of spanwise constant PVS frequency is therefore applied herein in a pessimistic aspect, although it is noted that the signature of PVS noise was observed by [19] even when PVS was confined to ~10% span near the tip of a single blade.
- (b)
- The latter is presumed to provoke blade vibration, if the frequency of PVS coincides with the frequency of the first bending mode of blade vibration.

#### 4.1. Aerodynamics: Blade Design

_{b}) = constant. The tendency toward keeping the chord constant along the span in the blade design is supported by the literature examples of [19,20]. Based on the well-known Cordier diagram [46] for turbomachines of favorable efficiency, axial flow fans have the specific diameter and the specific speed within the approximate ranges of 1 ≤ δ ≤ 1.5 and 2 ≤ σ ≤ 3, respectively. These ranges correspond to the global total pressure rise coefficient and flow coefficient within the ranges of 0.05 ≤ Ψ

_{t}≤ 0.25 and 0.1 ≤ Φ ≤ 0.5, respectively. Thus, values in Table 3 fit to the Cordier diagram well. Regulation 327/2011/EU5 issued energy efficiency requirements regarding fans in the EU [47], driven by motors with an electric input power between 125 W and 500 kW. According to this, the target total efficiency of an axial fan is 0.5 ≤ η

_{t}≤ 0.6 depending on the arrangement and input power. Considering a more economical operation, we are moving toward higher efficiency levels; therefore, η

_{t}≈ 0.7 is chosen.

_{x}is the axial velocity component; u is the circumferential velocity; and Δc

_{u}is the increase of tangential velocity due to the rotor.

_{x}(r

_{b})/u

_{tip}is the local flow coefficient; R is the dimensionless radius; and ψ

_{t,is}is the local isentropic total pressure rise coefficient. As a brief approximation, the circumferential velocity u, representing a solid body rotation, dominates in U

_{0}. Therefore, U

_{0}tends to approximately linearly increase with R. In a refined calculation, presented later, c

_{x}/u

_{tip}and Δc

_{u}/u

_{tip}are taken into account when obtaining U

_{0}.

_{PVS}, b(r

_{b}) tends to increase along the span by such means that its increase matches with the spanwise increase of U

_{0}.

_{D}. Based on Figure 4, C

_{D}(R

_{mid}) = 0.032 is chosen, which corresponds to α = 6.8°. In the design of the PVS-affected rotor, the following range of C

_{D}(R) was used: 0.0196 ≤ C

_{D}≤ 0.0410. Furthermore, as Figure 4 illustrates, the C

_{D}data within the design range are assigned to α data, and via such assignment, they also determine the design range for the local lift coefficient C

_{L}(α). Therefore, the lift-to-drag ratio LDR(α) = C

_{L}/C

_{D}data are also obtained for the entire design range, incorporating the data at R

_{mid}. Thus, each of C

_{D}(R

_{mid}), C

_{L}(R

_{mid}), and LDR(R

_{mid}) are available; these quantities will play an important role in the further investigation of the PVS-affected rotor, as presented later.

#### Iterative Method

_{PVS}, the nearly linear spanwise increase of U

_{0}is matched in blade design with the spanwise increase of C

_{D}. To be able to design such a fan, an iterative method is elaborated as follows, using the data in Table 3 as the basis. Using Equation (16) as an initial guess, the increase of tangential velocity due to the rotor is neglected; therefore, W(R) is calculated from only the local flow coefficient φ = c

_{x}(r

_{b})/u

_{tip}and the dimensionless radius corresponding to the rigid body rotation. In the present methodology, uniform axial inlet condition is assumed, c

_{x}(r

_{b})/u

_{tip}= constant. Furthermore, based on [46] as an approximation, the change of meridional—i.e., axial—velocity is neglected through the rotor. The above implies that spanwise constant axial velocity is presumed, as a brief approximation.

_{mid})/C

_{D}(R

_{mid}), the drag coefficients are determined along the blade span. As described earlier, the C

_{D}(R), α(R) and C

_{L}(R) data are assigned to each other, via Figure 4. As a next stage of the design, Δc

_{u}/u

_{tip}is expressed from the simplified work equation of an elemental rotor:

_{b}π/N is the blade spacing. With the knowledge of Δc

_{u}(R)/u

_{tip}, the local isentropic total pressure rise coefficient ψ

_{t,is}(R) is expressed from the Euler equation of turbomachines:

_{t,is}is the isentropic total pressure rise, and ρ

_{a}is the density of air. This computed ψ

_{t,is}(R) is then substituted into Equation (16) for calculating a new approximation of W(R) in the next iteration loop. Fast convergence is obtained in two to three iteration loops. The results are judged to be converged if the relative difference in ψ

_{t,is}(R) for the consecutive iteration steps becomes less than 2%.

_{t,is}(R) and α(R) are actually the results of the design process. The spanwise distributions of the calculated quantities are shown in Figure 10, where γ(R) is stagger angle measured from the circumferential direction. Annulus-averaging of ψ

_{t,is}(R) represents the global isentropic total pressure rise obtained as Ψ

_{t}/η

_{t}, using the data in Table 3.

#### 4.2. Mechanically or Acoustically Unfavorable Design Cases

_{0}and b in Equation (14). However, b was previously expressed, as shown in Equation (17). Therefore, it is only necessary to deal with U

_{0}. Based on Equations (20) and (21), the free-stream velocity is written as follows:

_{b})/u(r

_{b}) = (2 r

_{b}π/N)/(2 r

_{b}πn) = 1/(nN) and C

_{D}= C

_{L}/LDR, Equation (23) is written as follows:

_{b}radius can be replaced by the same characteristic taken at R

_{mid}. Furthermore, as a brief approximation η

_{t}(r

_{b}) = η

_{t}(R

_{mid}) ≈ constant is presumed. With respect to the foregoing, the first term on the right-hand side of Equation (25) can be expressed:

_{tip}= D

_{tip}πn is the tip circumferential speed. The second and the third terms are written as follows:

_{tip}/a

_{b}velocity ratio for which PVS results in blade resonance, if the nondimensional characteristics—valid for an entire PVS-affected rotor family under survey—are substituted into the right-hand side of the equation. With knowledge of the blade material, a

_{b}can be obtained (cf. Equation (10) and the paragraph below), and thus, the critical u

_{tip}value can be computed. Hence, critical rotor diameter X rotor speed data couples can be discovered for an entire rotor family, consisting of rotors of various diameters and speeds.

_{tip}value can be compared to the aforementioned critical one. Thus, it can be judged whether a risk of blade resonance may occur by changing the rotor speed, e.g., via a frequency converter. Furthermore, if the rotor diameter and the rotor speed are fixed, all dimensional quantities can be calculated, with the knowledge of nondimensional data in Table 3 as well as on the right-hand side of Equation (25). This makes possible the calculation of f

_{PVS}, using Equation (25), for acoustics evaluation. By such means, the third-octave band incorporating PVS can be identified and critically evaluated. For this purpose, the A-weighting graph [49] is to be considered. The plateau of the A-weighting graph represents the most sensitive part of the human audition. Keeping f

_{PVS}away from this plateau in blade design, by selecting the appropriate operational and geometrical characteristics, gives a potential for moderating the impact of fan noise on humans. If such design intent cannot be realized for modifying f

_{PVS}, the PVS phenomenon in itself is to be suppressed, necessitating modifications in the blade layout, e.g., boundary-layer tripping. However, such modifications are to be treated with criticism, as, e.g., boundary layer tripping may undermine the performance of the fan [3]. Such undesired effects justify the present intent by the authors to accept the occurrence of PVS but “mistune” it toward uncritical frequencies by simple preliminary design means, as a first approach. Such mistuning, being beneficial from both vibration and noise points of view, is to be performed in the preliminary blade design by the negation—i.e., avoidance—of the worst-case design and operational scenarios represented by PVS-affected rotors. For this reason, setting up guidelines for the worst-case design scenarios is of practical value.

## 5. Calculation Example for the Designed Rotor

_{b}in terms of the calculation process. The calculated values are summarized in Table 4.

_{PVS}and f

_{B,1}[Equation (10)] values for the presented case study. Furthermore, the lower the tip speed, the lower the fluctuating force causing vibration. However, the risk of blade vibration cannot be excluded for other design cases, characterized by modified data in Table 3 and for other—i.e., higher-order bending, as well as torsional—modes of vibration. Therefore, an important future task—as part of the ongoing research project—is to systematically explore risky cases (operational, geometrical, and material characteristics) from the resonance point of view. The methodology presented herein can be generally applied for such systematic studies.

_{tip}= 0.900 m, n = 1450 1/min. Such values are relevant in industrial ventilation. They result in u

_{tip}= 68.3 m/s. Considering data in Table 4 as well as previously fixed further parameters, the additional quantities required to calculate the PVS frequency, according to Equation (28), are derived. The values of the computed quantities are presented in Table 5.

_{B,1}= 116 Hz. For the sake of completeness, the following mandatory engineering investigation is to be performed. On the one hand, is to be checked whether the computed eigenfrequency is sufficiently far from the nominal rotational frequency. The rotational speed n = 1450 1/min, which corresponds to 24 Hz, is thus ≈ 20% of the first bending eigenfrequency, which is satisfactory.

_{B,1}≈ 5 × 24 Hz = 120 Hz, the critical element number is five, which should be avoided by all means. For example, upon demand, the application of three fan-supporting struts upstream of the rotor fulfills this condition.

## 6. Conclusions and Future Remarks

- (a)
- The critical tip speed that may cause resonance can be estimated for various blade materials, according to Equation (31).
- (b)
- On the basis of (a), the critical rotor speed n can be calculated for axial fans of known diameter. Thus, it can be judged whether a risk of blade resonance may occur by changing the rotor speed.
- (c)
- The expected PVS frequency can be determined by knowing the rotor speed of the fan. Therefore, the adverse acoustic effect of PVS can be forecasted on the basis of the A-weighting graph.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

Latin Letters | |

A | Cross-section [m^{2}] |

a_{b} | Wave propagation speed = (E/ρ_{b})^{1/2} [m/s] |

AR | Aspect ratio = l/c [-] |

b | Transversal dist. between vortex rows [m] |

c/s | Blade solidity [-] |

D | Rotor diameter [m] |

d_{TE} | Trailing edge thickness [m] |

E | Young modulus [Pa] |

F | Characteristic function |

f | Dominant frequency [Hz] |

g | Lateral displacement [m] |

G | Shear modulus [Pa] |

h | Maximum height of the camber line [m] |

h/c | Relative camber [-] |

i | Order number, e.g., 1, 2, 3 etc. |

I | Second moment of area [m^{4}] |

I_{p} | Polar moment of area [m^{4}] |

I_{t} | Torsional constant [m^{4}] |

K* | Empirical coefficient ≈ 1.2 [-] |

K_{1} | Preliminary design constant [-] |

K_{b} | Blade mechanics coefficient [-] |

K_{t} | Mechanical constant [-] |

K_{B,i} | Bending mode constant [-] |

c | Blade chord length [m] |

c_{x} | Axial velocity component [m/s] |

C_{D} | Drag coefficient [-] |

C_{L} | Lift coefficient [-] |

l | Blade span [m] |

LDR | Lift-to-drag ratio = C_{L}/C_{D} [-] |

N | Blade count [-] |

n | Rotor speed [1/s] |

R | Dimensionless radius = r/r_{tip} [-] |

r | Radial coordinate [m] |

Re_{c} | Reynolds number = c U_{0} /ν_{a} [-] |

s | Blade spacing = 2rπ/N [m] |

St | Strouhal number [-] |

St* | Universal Strouhal number [-] |

t | Profile thickness [m] |

t/c | Relative thickness [-] |

u | Rotor circumferential velocity [m/s] |

U_{0} | Free-stream velocity = (w_{1} + w_{2})/2 [m/s] |

U | Length of midwall perimeter [m] |

w | Relative velocity component [m/s] |

W | Dimensionless free-stream velocity [-] |

v | Absolute velocity component [m/s] |

x, y, z | Cartesian coordinates [m] |

Greek Letters | |

α | Angle of attack [°] |

β_{i}l | Vibration constant ≈ (2i − 1)π/2 |

γ | Stagger angle [°] |

δ | Specific diameter [-] |

ζ | Rotation angle [rad] |

Δc_{u} | Increase of tangential velocity [m/s] |

Δp_{t,is} | Isentropic total pressure rise [Pa] |

η | Efficiency [-] |

Θ | Momentum thickness of blade wake [m] |

ν | Hub-to-tip ratio [-] |

ν_{a} | Kinematic viscosity of air [m^{2}/s] |

ν_{P} | Poisson’s ratio [-] |

ρ | Density [kg/m^{3}] |

σ | Specific speed [-] |

τ | Time [s] |

φ | Local flow coefficient [-] |

Φ | Global flow coefficient [-] |

ψ | Local pressure rise coefficient [-] |

Ψ | Global pressure rise coefficient [-] |

Subscripts and Superscripts | |

1 | Rotor inlet |

2 | Rotor outlet |

a | Air |

b | Blade |

B | Bending |

crit | Critical |

i | Order-number i.e., 1, 2, 3 |

is | Isentropic |

mid | Mid-span position |

n | Order-number i.e., 1, 2, 3 |

PVS | Profile vortex shedding |

t | Total |

T | Torsional |

TE | Trailing edge |

tip | Blade tip |

Abbreviations | |

2D | Two-dimensional |

3D | Three-dimensional |

B | Bending mode |

CG | Center of gravity |

CFD | Computational fluid dynamics |

PVS | Profile vortex shedding |

T | Torsional mode |

TE | Trailing edge |

VS | Vortex shedding |

FEM | Finite element method |

## Appendix A

Band No. | Lower Band Limit | Center Frequency | Upper Band Limit |
---|---|---|---|

(Hz) | (Hz) | (Hz) | |

1 | 11.2 | 12.5 | 14.1 |

2 | 14.1 | 16 | 17.8 |

3 | 17.8 | 20 | 22.4 |

4 | 22.4 | 25 | 28.2 |

5 | 28.2 | 31.5 | 35.5 |

6 | 35.5 | 40 | 44.7 |

7 | 44.7 | 50 | 56.2 |

8 | 56.2 | 63 | 70.8 |

9 | 70.8 | 80 | 89.1 |

10 | 89.1 | 100 | 112 |

11 | 112 | 125 | 141 |

12 | 141 | 160 | 178 |

13 | 178 | 200 | 224 |

14 | 224 | 250 | 282 |

15 | 282 | 315 | 355 |

16 | 355 | 400 | 447 |

17 | 447 | 500 | 562 |

18 | 562 | 630 | 708 |

19 | 708 | 800 | 891 |

20 | 891 | 1000 | 1122 |

21 | 1122 | 1250 | 1413 |

22 | 1413 | 1600 | 1778 |

23 | 1778 | 2000 | 2239 |

24 | 2239 | 2500 | 2818 |

25 | 2818 | 3150 | 3548 |

26 | 3548 | 4000 | 4467 |

27 | 4467 | 5000 | 5623 |

28 | 5623 | 6300 | 7079 |

29 | 7079 | 8000 | 8913 |

30 | 8913 | 10,000 | 11,220 |

31 | 11,220 | 12,500 | 14,130 |

32 | 14,130 | 16,000 | 17,780 |

33 | 17,780 | 20,000 | 22,390 |

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**Figure 4.**Lift and drag coefficients and lift-to-drag ratios as a function of angle of attack for 8% cambered plate at Re

_{c}= 3 × 10

^{5}measured by Wallis [44].

l | c | t | h | ρ_{b} | E | ν_{P} |
---|---|---|---|---|---|---|

264 mm | 120 mm | 2.40 mm | 9.60 mm | 7850 kg/m^{3} | 200 GPa | 0.30 |

Mode | f_{FEM} [Hz] | 1/3 Octave Band No. | f_{analytical} [Hz] | 1/3 Octave Band No. | Discrepancy [%] |
---|---|---|---|---|---|

B,1 | 116.5 | 11 | 116.1 | 11 | 0.3 |

B,2 | 595.2 | 18 | 727.6 | 19 | 18.2 |

B,3 | 1146.7 | 21 | 2037.8 | 23 | 43.7 |

T,1 | 140.1 | 11 | 117.4 | 11 | 19.4 |

T,2 | 472.8 | 17 | 352.1 | 15 | 34.3 |

T,3 | 949.6 | 20 | 586.9 | 18 | 61.8 |

c/D_{tip} | ν | AR | Ψ_{t} | Φ | N | η_{t} | δ | σ |
---|---|---|---|---|---|---|---|---|

0.133 | 0.415 | 2.22 | 0.143 | 0.309 | 5 | 0.700 | 1.11 | 2.39 |

Steel | Aluminum | Polycarbonate | ||
---|---|---|---|---|

a_{b} | [m/s] | 5000 | 5100 | 1350 |

u_{tip,crit} | [m/s] | 1.80 | 1.83 | 0.49 |

D_{tip} | c | C_{L} (R_{mid}) | LDR (R_{mid}) | s_{tip} | f_{PVS} |
---|---|---|---|---|---|

0.900 m | 0.120 m | 1.30 | 43.5 | 0.565 | 4520 Hz |

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**MDPI and ACS Style**

Daku, G.; Vad, J.
Preliminary Design Guidelines for Considering the Vibration and Noise of Low-Speed Axial Fans Due to Profile Vortex Shedding. *Int. J. Turbomach. Propuls. Power* **2022**, *7*, 2.
https://doi.org/10.3390/ijtpp7010002

**AMA Style**

Daku G, Vad J.
Preliminary Design Guidelines for Considering the Vibration and Noise of Low-Speed Axial Fans Due to Profile Vortex Shedding. *International Journal of Turbomachinery, Propulsion and Power*. 2022; 7(1):2.
https://doi.org/10.3390/ijtpp7010002

**Chicago/Turabian Style**

Daku, Gábor, and János Vad.
2022. "Preliminary Design Guidelines for Considering the Vibration and Noise of Low-Speed Axial Fans Due to Profile Vortex Shedding" *International Journal of Turbomachinery, Propulsion and Power* 7, no. 1: 2.
https://doi.org/10.3390/ijtpp7010002