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

Preliminary Design Guidelines for Considering the Vibration and Noise of Low-Speed Axial Fans Due... International Journal of Turbomachinery Propulsion and Power Article Preliminary Design Guidelines for Considering the Vibration and Noise of Low-Speed Axial Fans Due to Profile Vortex Shedding Gábor Daku * and János Vad Department of Fluid Mechanics, Faculty of Mechanical Engineering, Budapest University of Technology and Economics, Bertalan Lajos u. 4-6, H-1111 Budapest, Hungary; vad@ara.bme.hu * Correspondence: daku@ara.bme.hu † This paper is a revised version of our paper published in Proceedings of the European Turbomachinery Conference ETC14, Gdansk, Poland, 12–16 April 2021. Abstract: This paper presents a critical overview on worst-case design scenarios for which low-speed axial flow fans may exhibit an increased risk of blade resonance due to profile vortex shedding. To set up a design example, a circular-arc-cambered plate of 8% relative curvature is investigated in twofold approaches of blade mechanics and aerodynamics. For these purposes, the frequency of the first bending mode of a plate of arbitrary circular camber is expressed by modeling the fan blade as a cantilever beam. Furthermore, an iterative blade design method is developed for checking the risky scenarios for which spanwise and spatially coherent shed vortices, stimulating pronounced vibration and noise, may occur. Coupling these two approaches, cases for vortex-induced blade resonance are set up. Opposing this basis, design guidelines are elaborated upon for avoiding such resonance. Based on the approach presented herein, guidelines are also developed for moderating the annoyance due to the vortex shedding noise. Keywords: axial flow fan; blade vibration; low tip speed; preliminary fan design; vortex shedding Citation: Daku, G.; Vad, J. Preliminary Design Guidelines for Considering the Vibration and Noise 1. Introduction of Low-Speed Axial Fans Due to Vortex shedding (VS) from low-speed axial flow fan rotor blades has become of Profile Vortex Shedding. Int. J. Turbomach. Propuls. Power 2022, 7, 2. engineering relevance in the past decades. The VS phenomenon discussed in this paper— https://doi.org/10.3390/ijtpp7010002 termed herein as profile vortex shedding (PVS), and illustrated in Figure 1—is not to be confused with the trailing-edge-bluntness, VS, which takes place past the blunt trailing edge Received: 28 September 2021 (TE) of the blade profile, acting as the aft portion of a bluff body [1]. In the aforementioned Accepted: 4 January 2022 literature, PVS is referred to as a laminar-boundary-layer VS, because it can only occur Published: 7 January 2022 if the boundary layer is initially laminar at least over one side of the blade profile. In Publisher’s Note: MDPI stays neutral this case, the initially laminar boundary layer, being separated near or after mid-chord with regard to jurisdictional claims in position, reattaches in the vicinity of the TE—thus resulting in a separation bubble—and published maps and institutional affil- finally undergoes a laminar-to-turbulent transition. Moreover, the position and the size iations. of the formed separation bubble plays a key role in tonal PVS noise emission. Recently, Yakhina at al. [2] published a detailed investigation about tonal TE noise radiated by low Reynolds number airfoils. They observed that a precondition for tonal noise emission is the formed separation bubble being sufficiently close to the TE. PVS may occur within a Copyright: © 2022 by the authors. certain Reynolds number range. Based on the literature [3–5], a lower limit of Re = 5  10 Licensee MDPI, Basel, Switzerland. is assumed herein, while the upper limit is determined by the critical Reynolds number This article is an open access article of the natural laminar-to-turbulent transition. When PVS is discussed for low-speed fans, distributed under the terms and as in the present paper, incompressible flow is considered by implying a Mach number conditions of the Creative Commons of 0.3. Attribution (CC BY-NC-ND) license (https://creativecommons.org/ licenses/by-nc-nd/4.0/). Int. J. Turbomach. Propuls. Power 2022, 7, 2. https://doi.org/10.3390/ijtpp7010002 https://www.mdpi.com/journal/ijtpp Int. J. Turbomach. Propuls. Power 2022, 7, 2 2 of 23 Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 2 of 23 Figure 1. Profile vortex shedding (PVS). Figure 1. Profile vortex shedding (PVS). V V arious arious model modelssar are e available available in in the li the literatur terature on the PVS phenomenon. In th e on the PVS phenomenon. In this is paper, paper, only only the c the classic lassic model by model by Tam Tam [6] [6] and and Wr Wright ight [[7] is re 7] is referr ferred ed to, in to, in orde orderr to to pr provide ovide aa straightforward and comprehensive interpretation on the mechanism. According to this straightforward and comprehensive interpretation on the mechanism. According to this model, PVS is related to a self-excited feedback loop. Due to the unstable laminar boundary model, PVS is related to a self-excited feedback loop. Due to the unstable laminar bound- layer, Tollmien–Schlichting instability waves are generated; these waves travel downstream ary layer, Tollmien–Schlichting instability waves are generated; these waves travel down- toward the TE where sound scattering occurs, and acoustic waves are created. The acoustic stream toward the TE where sound scattering occurs, and acoustic waves are created. The waves propagate upstream to amplify the original instabilities. If appropriate phase acoustic waves propagate upstream to amplify the original instabilities. If appropriate conditions are fulfilled, the disturbances are amplified at some frequencies, thus closing the phase conditions are fulfilled, the disturbances are amplified at some frequencies, thus feedback loop. Later, a number of authors, e.g., Nash et al. [8], carried out a critical revision closing the feedback loop. Later, a number of authors, e.g., Nash et al. [8], carried out a on the aforementioned feedback loop model. PVS may generate vibration on the blade. critical revision on the aforementioned feedback loop model. PVS may generate vibration As the studies by Ausoni et al. [9] suggest, the mechanisms of periodic vortex shedding on the blade. As the studies by Ausoni et al. [9] suggest, the mechanisms of periodic vortex and periodic blade vibration may mutually be coupled at a blade eigenfrequency, within a shedding and periodic blade vibration may mutually be coupled at a blade eigenfre- “lock-in” phenomenon. quency, within a “lock-in” phenomenon. In the case of low-speed axial flow fan blades, the difference between PVS and TE- In the case of low-speed axial flow fan blades, the difference between PVS and TE- bluntness VS in their physical mechanisms manifests itself in scaling techniques and the bluntness VS in their physical mechanisms manifests itself in scaling techniques and the values of the Strouhal number being also different. values of the Strouhal number being also different. For blade profiles with thick or blunt trailing edges, the TE-bluntness vortex shed- For blade profiles with thick or blunt trailing edges, the TE-bluntness vortex shed- ding [10] can be characterized by the Strouhal number based on the free-stream velocity U ding [10] can be characterized by the Strouhal number based on the free-stream velocity and the TE thickness d : TE U0 and the TE thickness dTE: St = f d /U = 0.20 (1) TE TE TE St = f d / U ≅ 0.20 TE TE TE 0 (1) For PVS [11,12], which is associated with the boundary layer transition and feedback mechanism: For PVS [11,12], which is associated with the boundary layer transition and feedback St = f b/U = 0.18 (2) PVS 0 mechanism: where f is the dominant frequency of the two types of VS, and b is the distance between St = f b / U ≅ 0.18 (2) PVS 0 the vortex rows. Yarusevych et al. [11] found that St* is universally valid for symmetrical, where f is the dominant frequency of the two types of VS, and b is the distance between relatively thick NACA airfoils for certain ranges of the Reynolds number and angle of attack, the vortex rows. Yar suggesting the usevy appellation ch et al. [11] of afound that “universal” St* Str is un ouhal iver number sally valSt id fo *, as r sym specified metrica in l, Equation relatively(2). thiHowever ck NACA , these airfoi airfoils ls for cert area not in rwidely anges of used the Reyno in axial l fan ds number and application; ther angle eforo e,f the present authors have extended the proposed St* definition as follows. Systematic attack, suggesting the appellation of a “universal” Strouhal number St*, as specified in wind-tunnel Equation (2). experiments However, th wer ese a e performed irfoils are not on widely blade section used in models axial ftypical an applfor icat low-speed ion; there- axial fans, using a single-component hot-wire probe. Based on the measured f and b values, fore, the present authors have extended the proposed St* definition as follows. Systematic the authors also confirmed the validity of St* for asymmetrical profile geometries, such as wind-tunnel experiments were performed on blade section models typical for low-speed 8% cambered plate and RAF-6E profiles, in a quasi-2D experimental analysis [13]. Thus, axial fans, using a single-component hot-wire probe. Based on the measured f and b val- the available experimental database on PVS frequency [5] was extended. ues, the authors also confirmed the validity of St* for asymmetrical profile geometries, From an engineering point of view, the practical aspects of PVS are remarkable in two such as 8% cambered plate and RAF-6E profiles, in a quasi-2D experimental analysis [13]. ways: vibration and noise. On the one hand, PVS creates a periodically fluctuating force Thus, the available experimental database on PVS frequency [5] was extended. normal to the chord, increasing the risk of blade vibration. On the other hand, VS appears From an engineering point of view, the practical aspects of PVS are remarkable in as the primary source of the aeroacoustics noise of low-speed axial flow fans [14–16]. Hence, two ways: vibration and noise. On the one hand, PVS creates a periodically fluctuating it is an important engineering objective to check and possibly to control the vibration and force normal to the chord, increasing the risk of blade vibration. On the other hand, VS noise of low-speed axial fans due to the PVS that is already in the preliminary design appears as the primary source of the aeroacoustics noise of low-speed axial flow fans [14– phase. It is worth noting that the signatures of blade vibration and PVS-related noise may 16]. Hence, it is an important engineering objective to check and possibly to control the coincide in the frequency spectra. In [17], the vibrometer connected to the airfoil indicated vibration and noise of low-speed axial fans due to the PVS that is already in the prelimi- a vortex-induced vibration, the dominant frequency of which coincided with that of the nary design phase. It is worth noting that the signatures of blade vibration and PVS-re- far-field tone. lated noise may coincide in the frequency spectra. In [17], the vibrometer connected to the Measurements on PVS in the literature are mostly related to isolated and steady airfoil indicated a vortex-induced vibration, the dominant frequency of which coincided airfoils [2,11–13,17,18]. Only a few studies dealt with PVS in the case of rotating blades of with that of the far-field tone. asymmetrical profiles being characteristic for realistic axial fans. Longhouse [19] detected Int. J. Turbomach. Propuls. Power 2022, 7, 2 3 of 23 PVS noise on an axial fan of four cambered plate blades with a constant blade chord. Nevertheless, except for a small segment (~10%) of the span of one blade in near-tip region, the PVS noise was suppressed by aft-chord serrations. Furthermore, the spanwise variation of free-stream velocity tended to broaden the noise signature of PVS, thus acting against a remarkable, well-detectable tonal PVS character. Grosche and Stiewitt [20] examined a four-bladed propeller-type axial fan rotor with a moderate sweep and twist. They observed PVS noise at a  4 angle of attack, viewed in the rotating frame of reference near the blade tip, for three different Reynolds numbers based on the chord length (Re = 9  10 , 5 5 1.3  10 , 2.6  10 ). 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]. In order to moderate the harmful noise and vibration effects due to PVS, pessimistic design scenarios have systematically been discovered for which such harmful effects are pronounced. Furthermore, these design characteristics were coupled with rotor dynamics consideration. Based on this coupling, an exemplary unfavorable design case was set up in terms of both operational and geometrical characteristics, for which PVS demonstrates an increased risk of blade resonance. Taking the rotor of such an unfavorable design case—termed hereafter the PVS-affected rotor—as reference, approximate semi-empirical design guidelines can already be elaborated in the preliminary design phase for checking and possibly avoiding PVS-induced resonance. The design guidelines for noise reduction, reported in [13], have also been further developed toward a more detailed model, as illustrated in this paper. 2. Blade Vibration: An Overview Turbomachinery blade profiles are suggested to be modeled in preliminary analysis as simple cantilever beams [24–30], which means that all degrees of freedom of the blades at the blade root, i.e., where connected to the hub, are constrained. The pure bending— or, in other words, transversal or flexural—vibrations of a prismatic beam with uniform cross-section according to time (t) and the coordinate along the longitudinal direction of the beam (z) are described by the partial differential equation below, which is derived from the Euler-Bernoulli’s beam theory [26,31]: 4 2 ¶ g(z, t) ¶ g(z, t) EI + r A = 0 (3) 4 2 ¶x ¶t where g(z, t) is the lateral displacement along the axis perpendicular to the blade chord; E is the Young modulus; I is the second moment of area with respect to the axis being parallel to the chord and fitting to the center of gravity (CG) of the blade section; A is the cross-sectional area of the blade profile; and r 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: (cos b l + cosh b l) i i F (z) = (cos b z cosh b z) (sin b z sinhb z) (4) i i i i i (sin b l sinhb l) i i where i is the order-number, i.e., 1, 2, 3 etc.; F(z) is the characteristic function or the normal mode of the beam; l is the radial extension of blade from hub to tip, i.e., the blade span, and b l  (2i 1)/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. Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 4 of 23 for some bending modes, generated on the basis of Equation (4). The vertical axis repre- Int. J. Turbomach. Propuls. Power 2022 sents the normal mode , 7, 2 (i.e., dimensionless displacement) and the horizontal axis shows 4 of 23 the dimensionless length of the beam. Figure 2. Normal bending modes of the beam. Figure 2. Normal bending modes of the beam. The pure torsion of the cantilever beam is governed by the following differential The pure torsion of the cantilever beam is governed by the following differential Equation [32]: 2 2 Equation [32]: ¶ z(z, t) ¶ z(z, t) GI + r I = 0 (5) t b p 2 2 2 2 ¶x ¶t ∂ ζ() z,τ ∂ ζ() z,τ GI + ρ I = 0 (5) t b p 2 2 where z(z, t) is the rotation angle around the longitudinal direction of the beam; G is the ∂x ∂τ shear modulus; I is the torsional stiffness or torsional constant; and I is the polar moment t p where ζ(z, τ) is the rotation angle around the longitudinal direction of the beam; G is the of area of the blade section. shear modulus; It is the torsional stiffness or torsional constant; and Ip is the polar moment It is important to note that in practice, mixing of modes can occur, and accordingly, of area of the blade section. various researchers have executed studies on experimental and theoretical evaluations It is important to note that in practice, mixing of modes can occur, and accordingly, of flexural-torsional vibration analysis, taking into account the coupling of flexural and various researchers have executed studies on experimental and theoretical evaluations of torsional modes, e.g., [24,27,28,31,33,34]. The present authors have only dealt with pure flexural-torsional vibration analysis, taking into account the coupling of flexural and tor- bending and torsional vibration modes, keeping in mind the simplest possible analytical sional modes, e.g., [24,27,28,31,33,34]. The present authors have only dealt with pure description mode. bending and torsional vibration modes, keeping in mind the simplest possible analytical In the literature, basic concepts are available for the analytical treatment of the vibra- description mode. tion of beams/blades affected by centrifugal force field due to their rotation. Such concepts In the literature, basic concepts are available for the analytical treatment of the vibra- regard both untwisted [31] and twisted [24,28,29] geometries. As discussed in [25], the tion of beams/blades affected by centrifugal force field due to their rotation. Such concepts centrifugal force originating from the rotation of the blades has a stiffness-increasing effect, regard both untwisted [31] and twisted [24,28,29] geometries. As discussed in [25], the i.e., it tends to moderate the inclination of the blade to vibrate. At the present state of centrifugal force originating from the rotation of the blades has a stiffness-increasing ef- research, the authors neglect the mechanical effect of the centrifugal field, for the following fect, i.e., it ten reasons: (a)dThe s to moder centrifugal ate the inc field lin tends ation of to be the blad of moderate e to vibrate significance . At the presen in the low-speed t state of rese fan arch, t blades he discussed authors neg her lec ein; t the mecha and (b) n the icaintention l effect of t is he cent to make rifua ga pessimistic—i.e., l field, for the follsafety- ow- increasing—preliminary design approach via neglecting the stiffness-increasing trend due ing reasons: (a) The centrifugal field tends to be of moderate significance in the low-speed fan to bl the ades centrifugal discussed h field. erei Blade n; antwist d (b) t tends he int to ent reduce ion is t the o mak bending e a peigenfr essimist equencies, ic—i.e., sa as fet FEM y- computations (not presented herein) demonstrate. Therefore, for twisted blades, the critical increasing—preliminary design approach via neglecting the stiffness-increasing trend frequencies of excitation tend to be shifted toward lower values. At the present state of due to the centrifugal field. Blade twist tends to reduce the bending eigenfrequencies, as research, the authors neglect the mechanical effect of blade twist. The reasonability of such FEM computations (not presented herein) demonstrate. Therefore, for twisted blades, the neglect is commented on later on. The concerted review of the effects of centrifugal force critical frequencies of excitation tend to be shifted toward lower values. At the present field and blade twisting, or taking into account the mixing of the pure vibration modes, is state of research, the authors neglect the mechanical effect of blade twist. The reasonability planned to be the subject of future research. of such neglect is commented on later on. The concerted review of the effects of centrifugal Even mechanical or fluid mechanical excitations can unavoidably induce the vibration force field and blade twisting, or taking into account the mixing of the pure vibration of the axial flow fan blade to a certain extent, as well as being a source of vibration in the modes, is planned to be the subject of future research. structure on which it is installed. Such excitation effect may derive from the interaction of Even mechanical or fluid mechanical excitations can unavoidably induce the vibra- the fan blades with the wake developing behind the elements placed upstream of the rotor tion of the axial flow fan blade to a certain extent, as well as being a source of vibration in e.g., supporting struts, inlet guide vanes, or even from the discussed PVS phenomenon. the structure on which it is installed. Such excitation effect may derive from the interaction For instance, corresponding to the pressure rise, a steady mean lift force acts on the blade. of the fan blades with the wake developing behind the elements placed upstream of the However, the lift force also has a varying component e.g., the fluctuating force due to PVS. rotor e.g., supporting struts, inlet guide vanes, or even from the discussed PVS phenom- As detailed by [25,35], both forces produce a bending moment, resulting in vibration. If the enon. For instance, corresponding to the pressure rise, a steady mean lift force acts on the dominant frequency of PVS coincides with an eigenfrequency of the blade, resonance may blade. However, the lift force also has a varying component e.g., the fluctuating force due occur, and the intensity of the vibration can only be limited if the mechanical structure is stiff enough or damped sufficiently. Therefore, it is to be treated with special care in the case of fan blades made of cambered sheet metal plates because their eigenfrequency—due to the moderate inertia of the cross-section and the resulting lower stiffness—is lower compared to that of the profiled blades [25]. Int. J. Turbomach. Propuls. Power 2022, 7, 2 5 of 23 Even though fans commonly operate with mechanical stresses far below the capacity of their material—and thus the vibration of the fan blade due to resonance does not lead to the fracture of the mechanical structure—, in the presence of the corresponding stresses, which reach a maximum near the blade root, there can be a risk of fatigue fracture. These stresses combine with the centrifugal ones; therefore, a mixed type, alternating stress condition appears, which must be considered in the design of the fan. Furthermore, the rotor may become imbalanced due to the deformation patterns associated with each resonance frequency of the blade, forcing the shaft to bend, escalating the initial imbalance and bending. If the excitation is sufficiently intense, resulting in a vibration of large amplitude, one of the fan blades can rub into the duct walls leading to rotor imbalance or the breaking down of the blades. 2.1. Analytical Treatment 2.1.1. Bending Modes In order to provide a straightforward and comprehensive approach in preliminary design, avoiding any need for numerical computation at the present phase of research and as a solutions of Equation (3), the following analytically expressed eigenfrequencies are considered for the bending modes of an arbitrary prismatic rod, as specified in the literature [26]: K E I B,i f = (6) B,i l A r Namely, the simple analytical formula above can easily be extended to higher-order bending modes by substituting the appropriate K constant into Equation (6). The values B,i of K for the first three bending modes are K = 0.560, K = 3.506 and K = 9.819, B,i B,1 B,2 B,3 respectively. 2.1.2. Torsional Modes By solving the expression in Equation (5), the eigenfrequency for torsional modes can be obtained using the analytical formula e.g., in [26], as follows: i 0.5 G I f = (7) T,i 2l r I b p In Equation (7) every variable can be determined—except I —by utilizing the material and geometrical parameters of the blade. To calculate the I torsional constant for a flat plate, the following relation is given in [26]: I = K ct (8) t t where c is the chord length (width of the plate); t is the thickness (height of the plate); and K 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 [36–38], the torsional constant of a thin-walled open tube cross-section of uniform thickness can be expressed as: I = Ut (9) where U is the length of the midwall perimeter, shown dashed later in Figure 5. Int. J. Turbomach. Propuls. Power 2022, 7, 2 6 of 23 2.2. Finite Element Method (FEM) The FEM is a useful and generally accepted tool for solving engineering problems numerically, such as the modal analysis of rotor blades [39–41]. The desired mechanical characteristics can be computed by dividing an arbitrary mechanical structure into simple geometric shapes and defining the material properties and governing connections among these elements. In this study, the frequency analysis of a low speed axial fan blade is carried out using a commercial FEM software program ANSYS Mechanical APDL 2019 R3. 2.2.1. Geometry, Materials, and Elements In the present paper, the hub is assumed to be rigid in comparison to the fan blade, thus all degrees of freedom of the blades at the hub, i.e., at the blade root, are constrained. Therefore, only one segment of the axial flow fan composed of a single blade without a hub part was examined. In accordance with the later investigated PVS-affected rotor blade profile of the circular-arc-cambered plate of 8% relative curvature and the dimensional and dimensionless values of Tables 3 and 5, the necessary geometrical characteristics for 3D modeling are summarized in Table 1. The reason for choosing this blade profile geometry will be explained in detail later in Section 3.2. For the FEM case study, the fan blade is made of structural steel with a density of 7850 kg/m , a Young’s modulus of 200 GPa, and a Poisson’s ratio (n ) of 0.3. Table 1. Blade geometrical and material parameters for the FEM case study. l c t h r E n 264 mm 120 mm 2.40 mm 9.60 mm 7850 kg/m 200 GPa 0.30 According to the literature [30,39], SHELL 181 element is suitable for analyzing thin to moderately-thick shell structures related to turbomachinery blades, such as wind turbine and axial flow fan blades. Hence, this type of element is applied to model the fan blade with 124 nodes in the axial direction and 256 nodes in the radial direction. Basically, SHELL 181 is a four-node element with six degrees of freedom at each node: translations in the x, y, and z directions and rotations about the x, y, and z-axes. However, it should be mentioned that the same solution could be obtained by using the 20-node Hex20 (SOLID 186) element of Ansys Workbench 2019 R3. 2.2.2. Mesh Convergence A mesh convergence study was performed to assure the optimum mesh number in terms of computational accuracy. The first three bending (B) and torsional (T) eigenfrequen- cies of the fan blade are computed in several mesh sizes (the element number varies from 258 to 507,904). By increasing the number of elements, the first bending eigenfrequency increases slightly and then becomes nearly constant. For higher-order bending and the torsional mode, similar effects can be observed. As shown in Figure 3, 31,744 elements are appropriate to get a sufficiently accurate solution. Nevertheless, it should be mentioned that even a model with a lower number of elements is sufficient for the third-octave band prediction of the first bending eigenfrequency (see Appendix A). Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 7 of 23 Int. J. Turbomach. Propuls. Power 2022, 7, 2 7 of 23 Figure 3. Mesh convergence study. Figure 3. Mesh convergence study. 2.3. Comparison of the Results 2.3. Comparison of the Results In Table 2, the first three analytical bending (B) and torsion (T) eigenfrequencies In Table 2, the first three analytical bending (B) and torsion (T) eigenfrequencies were were calculated based on Equations (5), (7) and (8), and compared to the eigenfrequency calculated based on Equations (5), (7) and (8), and compared to the eigenfrequency ob- obtained by means of FEM. The first column indicates the vibration mode case under tained by means of FEM. The first column indicates the vibration mode case under dis- discussion. In the second and the fourth columns, FEM and the analytical eigenfrequencies cussion. In the second and the fourth columns, FEM and the analytical eigenfrequencies are summarized, respectively. In the third and the fifth columns, the appropriate band are summarized, respectively. In the third and the fifth columns, the appropriate band number of the third-octave band resolution is listed in accordance with Appendix A. The number of the third-octave band resolution is listed in accordance with Appendix A. The “Discrepancy” column contains the relative discrepancy of f in comparison to f . analytical FEM “Discrepancy” column contains the relative discrepancy of fanalytical in comparison to fFEM. Table 2. Geometrical and material parameters of the blade. Table 2. Geometrical and material parameters of the blade. f 1/3 Octave f 1/3 Octave Discrepancy FEM analytical fFEM 1/3 Octave Band fanalytical 1/3 Octave Band Discrepancy Mode Mode [Hz] Band No. [Hz] Band No. [%] [Hz] No. [Hz] No. [%] B,1 116.5 11 116.1 11 0.3 B,1 116.5 11 116.1 11 0.3 B,2 595.2 18 727.6 19 18.2 B,2 595.2 18 727.6 19 18.2 B,3 1146.7 21 2037.8 23 43.7 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,1 140.1 11 117.4 11 19.4 T,3 949.6 20 586.9 18 61.8 T,2 472.8 17 352.1 15 34.3 T,3 949.6 20 586.9 18 61.8 From Table 2, it can be observed that the analytical treatment for the first bending (B,1) eigenfrequency is in good agreement with the FEM result. Nevertheless, for the From Table 2, it can be observed that the analytical treatment for the first bending higher-order bending and the torsional modes, the relative discrepancy between analytical (B,1) eigenfrequency is in good agreement with the FEM result. Nevertheless, for the and FEM results becomes greater and tends to increase with the increasing of the order of higher-order bending and the torsional modes, the relative discrepancy between analyti- the modes. A possible explanation for this is the partial violation of the briefly presented cal and FEM results becomes greater and tends to increase with the increasing of the order Euler–Bernoulli thin beam theory. Hence, the AR of the beam is equal to 2.2; namely, the of the modes. A possible explanation for this is the partial violation of the briefly presented beam is not considered to be slender (AR > 10). Euler–Bernoulli thin beam theory. Hence, the AR of the beam is equal to 2.2; namely, the Although there have been alternative methodologies to obtain a more accurate an- beam is not considered to be slender (AR > 10). alytical prediction for short beams (AR < 10), e.g., Timoshenko’s beam theory [31], the Although there have been alternative methodologies to obtain a more accurate ana- present paper focuses on Euler–Bernoulli’s beam theory, due to the following reasons. lytical prediction for short beams (AR < 10), e.g., Timoshenko’s beam theory [31], the pre- (a) On the one hand, the authors aim to create a closed analytical formula with the most sent paper focuses on Euler–Bernoulli’s beam theory, due to the following reasons. (a) On straightforward possible analytical description in mind. (b) On the other hand, as shown the one hand, the authors aim to create a closed analytical formula with the most straight- in [31,42], the relative discrepancy between the analytical first bending eigenfrequencies, forward possible analytical description in mind. (b) On the other hand, as shown in calculated based on the two different theories, is less than 5–10%, which, fitting for point [31,42], the relative discrepancy between the analytical first bending eigenfrequencies, cal- (a), is considered to be an acceptable approximation. Based on this and the highlighted role culated based on the two different theories, is less than 5–10%, which, fitting for point (a), of the first bending mode detailed in Section 3.1, the confirmation of the analytical model is considered to be an acceptable approximation. Based on this and the highlighted role by FEM is of primary significance for the first bending mode only and is of secondary of the first bending mode detailed in Section 3.1, the confirmation of the analytical model importance for the other modes. by FEM is of primary significance for the first bending mode only and is of secondary Despite the simplification assumption discussed at the beginning of the chapter (un- importance for the other modes. twisted and non-rotating fan blade), supplementary FEM case studies were carried out Despite the simplification assumption discussed at the beginning of the chapter (un- for the twisted and rotating blade as well. The FEM results demonstrated that both the twisted and non-rotating fan blade), supplementary FEM case studies were carried out blade twisting and the presence of a centrifugal force field have a bending eigenfrequency- for the twisted and rotating blade as well. The FEM results demonstrated that both the reducing effect, providing an upper estimation for the first bending eigenfrequency of the blade twisting and the presence of a centrifugal force field have a bending eigenfrequency- blade. As the later calculation example illustrates, even in the case of the untwisted and Int. J. Turbomach. Propuls. Power 2022, 7, 2 8 of 23 stationary fan blade, only impractically low tip speeds would result in the coincidence of the first bending eigenfrequency and PVS frequency. Therefore, the simplified fan blade model applied by the present authors can be considered as a pessimistic design scenario. 3. Blade Mechanics: An Exemplary Case Study 3.1. The Importance of the First-Order Bending Mode The risk in fan operation due to blade vibration is simultaneously viewed in the present paper from the following two perspectives, which are related to each other. (a) The risk of instantaneous reduction of the gap between the blade tip and the casing. From the perspective of the tip gap reduction, the vibration modes exhibiting monotonously increasing deformation amplitude along the blade height are considered the riskiest. These are the first bending and first torsional modes. (b) The risk of instantaneous rotor imbalance. This occurs in cases when the weight point of the entire blade is displaced in the transversal direction from its original position, fitting on the (approximately) radial blade stacking line, for which the rotor has originally been balanced. From the perspective of rotor imbalance, none of the torsional modes are considered to be risky, as they tend to leave the weight point of the entire blade (approximately) in its original position. In addition, according to their wavy vibration pattern, higher-order bending modes are considered to exhibit only a moderate transversal displacement of the blade weight point. Therefore, among the various vibrational modes caused by PVS excitation, the first bending mode is judged to be the most critical, and the higher-order bending modes and torsional modes are considered of secondary significance. Consequently, out of the an- alytical solutions in Equation (5), the “first-order bending mode” is taken herein as an illustrative example of the analytical treatment elaborated upon by the authors. Neverthe- less, the reasonability of this choice is justified and supported by the subsequent examples, as follows. The impact of PVS on both noise and vibration is presumed by the present authors to be pronounced when extensive and coherent vortices are shed with uniform frequency along a dominant portion of the blade span. In this case, PVS is assumed to exhibit pressure fluctuations over the blade suction and pressure surfaces, causing chord-normal forces. Assuming spanwise and spatially coherent shed vortices, the resultant fluctuating forces are in phase over the entire span. Furthermore, PVS and the associated forces may occur farther upstream of the TE (cf. [12]). Such a PVS-induced excitation likely triggers the first-order bending mode. An upstream stator, e.g., the nearly radially aligned supporting struts located upstream of the rotor, is able to cause wake-blade—also known as “rotor-stator”—interaction, in which the upstream wakes of the stator are swept downstream into the axial flow fan blade- row. The wakes are parallel with the relative velocity (w), thus the interaction manifests itself as a nearly simultaneous, spatially coherent aerodynamic excitation along the entire blade span. Hence, the fan blades are acted by chord-normal fluctuating force, resulting in a spatially coherent bending moment rather than torsional. In addition, in [25] as a general approach, the frequency of the first bending mode of the blades is to be kept far away from the frequencies of excitation. As it was just mentioned, elements located upstream of the rotor—supporting struts and inlet guide vanes—cause rotor–stator interaction as aerodynamic excitation to the rotor blades, occur- ring at a frequency of rotational frequency multiplied by the number of upstream elements. On the other hand, such excitation effects correspond to rotor imbalance (“shaker effect”) appearing as mechanical excitation at the rotational frequency of the fan. Moreover, as illustrated in [5], the fluctuation of chord-normal force due to PVS may lead to a variance of the lift coefficient in the order of a magnitude of 10 percent of the temporal mean value. As demonstrated in [35], the varying component of lift force causes a bending moment on the blade. When extensive and coherent vortices are shed with uniform frequency along a dominant portion of the blade span, they represent spatially coherent elemental aerodynamic excitation forces, being in phase over the elemental blade Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 9 of 23 a bending moment on the blade. When extensive and coherent vortices are shed with uni- Int. J. Turbomach. Propuls. Power 2022, 7, 2 9 of 23 form frequency along a dominant portion of the blade span, they represent spatially co- herent elemental aerodynamic excitation forces, being in phase over the elemental blade sections, and thus, integrated into a pronounced overall bending moment. The phase sections, and thus, integrated into a pronounced overall bending moment. The phase identity of the elemental excitation forces along the blade span matches with the phase identity of the elemental excitation forces along the blade span matches with the phase identity of blade deformation in the first bending mode of the blade, if the frequency of identity of blade deformation in the first bending mode of the blade, if the frequency of PVS matches with the eigenfrequency related to the first bending mode. PVS matches with the eigenfrequency related to the first bending mode. 3.2. Eigenfrequency 3.2. Eigenfrequency In order to build up a straightforward model for the interaction of the blade and the In order to build up a straightforward model for the interaction of the blade and the fluctuating aerodynamic force due to PVS, some simplifications must be introduced. First fluctuating aerodynamic force due to PVS, some simplifications must be introduced. First of all, a circular-arc-cambered plate of 8% relative curvature is chosen as a blade profile of all, a circular-arc-cambered plate of 8% relative curvature is chosen as a blade profile for for several reasons: several reasons: (a) At moderate Reynolds numbers and angles of attack (α), the cambered plate pro- (a) At moderate Reynolds numbers and angles of attack (a), the cambered plate produces duces reasonably high CL, that is comparable with an airfoil profile, i.e., RAF-6E [43], reasonably high C , that is comparable with an airfoil profile, i.e., RAF-6E [43], thus thus enabling the design of blades of relatively high specific performance, i.e., utiliz- enabling the design of blades of relatively high specific performance, i.e., utilizing the ing the loading capability of the blade sections. loading capability of the blade sections. (b) At 8% relative curvature, the lift-to-drag (LDR) is near the maximum among the cam- (b) At 8% relative curvature, the lift-to-drag (LDR) is near the maximum among the bered plates of various relative camber, thus enabling the design for reasonably high cambered plates of various relative camber, thus enabling the design for reasonably efficiency [44]. high efficiency [44]. (c) (c) In accord In accordance ance with the with the afor afor ementioned p ementioned rpractical actical aspec aspects, ts, it is it a c is a amb camber ered plate o ed plate f 8% of relative camber for which hot-wire measurement data are made available by the pre- 8% relative camber for which hot-wire measurement data are made available by the sent authors present authors on P on VPVS S at different at different free free-str -stream eam ve velocities locities and ang and angles les of at of attack tack [ [1 13 3]. ]. As As per per the illustration, the illustration, F Figur igure 4 shows e 4 shows C CL, , C CD, and , and LD LDR R va values lues a as s a fu a function nction of the of the angl angle e L D of attack for the 8% cambered plate. of attack for the 8% cambered plate. Figure 4. Lift and drag coefficients and lift-to-drag ratios as a function of angle of attack for 8% Figure 4. Lift and drag coefficients and lift- 5 to-drag ratios as a function of angle of attack for 8% cambered plate at Re = 3  10 measured by Wallis [44]. cambered plate at Rec = 3 × 10 measured by Wallis [44]. Secondly, the fan blade is presumed to consist of geometrically identical blade sections Secondly, the fan blade is presumed to consist of geometrically identical blade sec- along the full span. This means that the blade chord c, plate thickness t, and height of the tions along the full span. This means that the blade chord c, plate thickness t, and height camber line h are constants (Figure 5). The measurement data presented by [44] and shown of the camber line h are constants (Figure 5). The measurement data presented by [44] and in Figure 4, are related to a relative thickness (t/c) of 2%. This value is representative in fan shown in Figure 4, are related to a relative thickness (t/c) of 2%. This value is representa- manufacturing; therefore, t/c is fixed at 2% for the present investigations. tive in fan manufacturing; therefore, t/c is fixed at 2% for the present investigations. Int. J. Turbomach. Propuls. Power 2022, 7, 2 10 of 23 Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 10 of 23 Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 10 of 23 Figure 5. Circular-arc-cambered plate (8% relative camber). Figure 5. Circular-arc-cambered plate (8% relative camber). Figure 5. Circular-arc-cambered plate (8% relative camber). Finally, as discussed earlier, it is assumed that all degrees of freedom of the blades at Finally, as discussed earlier, it is assumed that all degrees of freedom of the blades at Finally, as discussed earlier, it is assumed that all degrees of freedom of the blades at the hub, i.e., at the blade root, are constrained. This is a reasonable approximation; e.g., the hub, i.e., at the blade root, are constrained. This is a reasonable approximation; e.g., if the hub, i.e., at the blade root, are constrained. This is a reasonable approximation; e.g., if if sheet metal blades connect to the hub with a welded joint, or, in the case of polymer sheet metal blades connect to the hub with a welded joint, or, in the case of polymer ma- sheet metal blades connect to the hub with a welded joint, or, in the case of polymer ma- material, if the entirety of the hub and blading assembly is injection-molded as a single terial, if the entirety of the hub and blading assembly is injection-molded as a single prod- terial, if the entirety of the hub and blading assembly is injection-molded as a single prod- product. The simplifications above enable us to model the blade as a cantilever beam uct. The simplifications above enable us to model the blade as a cantilever beam (or uct. The simplifications above enable us to model the blade as a cantilever beam (or (or clamped beam) subjected to free vibration. In accordance with Section 3.1, the lowest clamped beam) subjected to free vibration. In accordance with Section 3.1, the lowest ei- clamped beam) subjected to free vibration. In accordance with Section 3.1, the lowest ei- eigenfrequency, related to the first-order bending mode, can be calculated as follows: genfrequency, related to the first-order bending mode, can be calculated as follows: genfrequency, related to the first-order bending mode, can be calculated as follows: 0.56 I E 0.56 I E 0.56 I E f = (10) B,1 f = f = B,1 (10) 2 2 (10) B,1 2 l A r l A ρ b l A ρ The The la last st term term on on the right- the right-hand hand side side of of Eq Equation uation (10), as (10), as the the squar square e r root of the sp oot of the specific ecific The last term on the right-hand side of Equation (10), as the square root of the specific modulus modulus E E/ /ρrb, is termed h , is termed her erein the w ein the wave ave pr propagation opagation speed, an speed, and d denoted denoted as as aab.. This This is is modulus E/ρb b, is termed herein the wave propagation speed, and denoted as a bb. This is actually the acoustic wave propagation speed in a long one-dimensional fictitious beam actually the acoustic wave propagation speed in a long one-dimensional fictitious beam 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. made of the blade material. The resultant swinging pattern is shown in Figure 6. made of the blade material. The resultant swinging pattern is shown in Figure 6. Figure 6. Cantilever beam subjected to free vibration (perpendicular to the blade chord). Figure 6. Cantilever beam subjected to free vibration (perpendicular to the blade chord). Figure 6. Cantilever beam subjected to free vibration (perpendicular to the blade chord). 3.3. Second Moment of Area 3.3. Second Moment of Area 3.3. Second Moment of Area The purpose of the present section is to demonstrate how the I/A term can be related The purpose of the present section is to demonstrate how the I/A term can be related The purpose of the present section is to demonstrate how the I/A term can be related to the basic geometrical characteristics of the circular-arc-cambered plate blade (c, h/c, to the basic geometrical characteristics of the circular-arc-cambered plate blade (c, h/c, t/c), to the basic geometrical characteristics of the circular-arc-cambered plate blade (c, h/c, t/c), t/c), creating a direct relationship between the first eigenfrequency and the geometrical creating a direct relationship between the first eigenfrequency and the geometrical param- creating a direct relationship between the first eigenfrequency and the geometrical param- parameters of the blade. However, no closed analytical relationship exists in the literature eters of the blade. However, no closed analytical relationship exists in the literature for eters of the blade. However, no closed analytical relationship exists in the literature for for such purpose. Therefore, an alternative method must be found to express the second such purpose. Therefore, an alternative method must be found to express the second term such purpose. Therefore, an alternative method must be found to express the second term term of the right-hand side of Equation (10) with the use of blade geometrical parameters, of the right-hand side of Equation (10) with the use of blade geometrical parameters, such of the right-hand side of Equation (10) with the use of blade geometrical parameters, such such as c, h, and t. As a first step, the second moment of area of a cambered plate and a flat as c, h, and t. As a first step, the second moment of area of a cambered plate and a flat plate as c, h, and t. As a first step, the second moment of area of a cambered plate and a flat plate plate with same geometrical parameter (t, c) were compared to each other. with same geometrical parameter (t, c) were compared to each other. with same geometrical parameter (t, c) were compared to each other. The quotient of the second moment of area and the cross-section, in case of a flat The quotient of the second moment of area and the cross-section, in case of a flat plate The quotient of the second moment of area and the cross-section, in case of a flat plate plate is: is: is: 3 2 (I /A) = (ct /12)/(ct) = t /12 (11) flat 3 2 3 2 () I A = (ct /12) (ct) = t 12 (11) () I A = (ct /12) (ct) = t 12 (11) flat For a cambered plate it is: flat For a cambered plate it is: For a cambered plate it is: (I /A) = K (I /A) , where K = K (t/c, h/c) (12) 1 1 1 cambered flat () I A = K() I A , where K = K (t/c, h/c) (12) () I A = K() I A , where K = K (t/c, h/c) 1 1 1 (12) cambered flat cambered 1 flat 1 1 I was obtained for the cambered plate-section using an analytical integration cambered process known from basic solid-state mechanics [45]. As background information for the Icambered was obtained for the cambered plate-section using an analytical integration Icambered was obtained for the cambered plate-section using an analytical integration reader, the values of K are presented in Figure 7 for representative relative thickness and process known from basic solid-state mechanics [45]. As background information for the process known from basic solid-state mechanics [45]. As background information for the relative camber (h/c) values. For the fitted curves in Figure 7, K was calculated for fixed t/c reader, the values of K1 are presented in Figure 7 for representative relative thickness and reader, the values of K1 are presented in Figure 7 for representative relative thickness and values for uniform steps of 0.01 h/c over the entire h/c range. In the blade design presented relative camber (h/c) values. For the fitted curves in Figure 7, K1 was calculated for fixed relative camber (h/c) values. For the fitted curves in Figure 7, K1 was calculated for fixed later, K (t/c = 0.02; h/c = 0.08) = 18.07 was used, in accordance with the previously selected t/c values for uniform steps of 0.01 h/c over the entire h/c range. In the blade design pre- t/c values for uniform steps of 0.01 h/c over the entire h/c range. In the blade design pre- blade geometrical parameters. sented later, K1 (t/c = 0.02; h/c = 0.08) = 18.07 was used, in accordance with the previously sented later, K1 (t/c = 0.02; h/c = 0.08) = 18.07 was used, in accordance with the previously selected blade geometrical parameters. selected blade geometrical parameters. Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 11 of 23 Substituting Equations (11) and (12) into Equation (10), the first bending eigenfre- quency can be expressed as follows: 0.56 t 0.56 (13) f = K a = K t a B,1 1 b b b 2 2 l 12 l Int. J. Turbomach. Propuls. Power 2022, 7, 2 11 of 23 where Kb = Kb (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. Figure Figure 7. 7. The The v value alue of of K K1 as a f as a function unction of relat of relative ive camber camber for for various various relative thicknesses. relative thicknesses. Substituting Equations (11) and (12) into Equation (10), the first bending eigenfre- 4. PVS-Affected Rotor: An Exemplary Case Study quency can be expressed as follows: The aim of this section is to outline a design case study, resulting in a rotor suspected to be unfavorable in terms of noise and vibration due to PVS. Although several unfavor- 0.56 t 0.56 able design cases of such kind can be inte f = K ntionaally genera = K ted, thi t a s paper intends to present (13) B,1 1 b b b 2 2 l l a single, representative design example. The resultant rotor will be termed herein the PVS- where K = K (t/c = 0.02; h/c = 0.08) = 1.23 is the blade mechanics coefficient. Case studies affected rotor. Such a design aims to serve as a basis for the realization of a case study b b considering other t/c and h/c values can be carried out using Figure 7. rotor that is expected to exhibit remarkable and experimentally well-detectable narrow- band signatures of PVS. By such means, the orders of the magnitude of harmfulness of 4. PVS-Affected Rotor: An Exemplary Case Study PVS-related effects are to be quantified in future experiments on the PVS-affected rotor, The aim of this section is to outline a design case study, resulting in a rotor suspected in comparison with other rotors of comparative design. The features of the PVS-affected to be unfavorable in terms of noise and vibration due to PVS. Although several unfavorable rotor are as follows: design cases of such kind can be intentionally generated, this paper intends to present (a) The rotor blading tends to exhibit a PVS of spanwise constant frequency, and thus, is a single, representative design example. The resultant rotor will be termed herein the theoretically presumed to realize large-scale, spatially coherent vortices over the PVS-affected rotor. Such a design aims to serve as a basis for the realization of a case dominant portion of the blade span. Such coherent vortices are assumed to cause study rotor that is expected to exhibit remarkable and experimentally well-detectable spatially correlated, narrowband noise, as well as mechanical excitation over the narrowband signatures of PVS. By such means, the orders of the magnitude of harmfulness dominant part of span at a given frequency. The condition of spanwise constant PVS of PVS-related effects are to be quantified in future experiments on the PVS-affected rotor, frequency is therefore applied herein in a pessimistic aspect, although it is noted that in comparison with other rotors of comparative design. The features of the PVS-affected the signature of PVS noise was observed by [19] even when PVS was confined to rotor are as follows: ~10% span near the tip of a single blade. (a) The rotor blading tends to exhibit a PVS of spanwise constant frequency, and thus, (b) The latter is presumed to provoke blade vibration, if the frequency of PVS coincides is theoretically presumed to realize large-scale, spatially coherent vortices over the with the frequency of the first bending mode of blade vibration. dominant portion of the blade span. Such coherent vortices are assumed to cause spatially correlated, narrowband noise, as well as mechanical excitation over the 4.1. Aerodynamics: Blade Design dominant part of span at a given frequency. The condition of spanwise constant PVS To be able to design a fan for which the spanwise constancy of PVS frequency is ful- frequency is therefore applied herein in a pessimistic aspect, although it is noted that filled—thus satisfying Equation (14), presented later—first, typical dimensionless design the signature of PVS noise was observed by [19] even when PVS was confined to ~10% and geometrical characteristics, representative of propeller-type fans, were systematically span near the tip of a single blade. gathered and summarized in Table 3. Here, the authors stress that, in order to guarantee (b) The latter is presumed to provoke blade vibration, if the frequency of PVS coincides the validity of Equation (9), the blade chord along the blade span was fixed, so c(rb) = with the frequency of the first bending mode of blade vibration. 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 Cor- 4.1. Aerodynamics: Blade Design dier diagram [46] for turbomachines of favorable efficiency, axial flow fans have the spe- To be able to design a fan for which the spanwise constancy of PVS frequency is cific diameter and the specific speed within the approximate ranges of 1 ≤ δ ≤ 1.5 and 2 ≤ fulfilled—thus satisfying Equation (14), presented later—first, typical dimensionless design σ ≤ 3, respectively. These ranges correspond to the global total pressure rise coefficient and geometrical characteristics, representative of propeller-type fans, were systematically and flow coefficient within the ranges of 0.05 ≤ Ψt ≤ 0.25 and 0.1 ≤ Φ ≤ 0.5, respectively. gathered and summarized in Table 3. Here, the authors stress that, in order to guarantee the validity of Equation (9), the blade chord along the blade span was fixed, so c(r ) = 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  d  1.5 and 2  s  3, respectively. These ranges correspond to the global total pressure rise coefficient and flow coefficient within the ranges of 0.05  Y  0.25 and 0.1  F  0.5, respectively. Thus, values in Table 3 fit to the Cordier diagram well. Regulation 327/2011/EU5 issued energy Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 12 of 23 Int. J. Turbomach. Propuls. Power 2022, 7, 2 12 of 23 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 effi- efficiency requirements regarding fans in the EU [47], driven by motors with an electric ciency of an axial fan is 0.5 ≤ ηt ≤ 0.6 depending on the arrangement and input power. input power between 125 W and 500 kW. According to this, the target total efficiency of an Considering a more economical operation, we are moving toward higher efficiency levels; axial fan is 0.5  h  0.6 depending on the arrangement and input power. Considering therefore, ηt ≈ 0.7 is chosen. a more economical operation, we are moving toward higher efficiency levels; therefore, h  0.7 is chosen. Table 3. Geometrical and material parameters of the blade for the PVS-affected rotor. Table 3. Geometrical and material parameters of the blade for the PVS-affected rotor. c/Dtip ν AR Ψt Φ N ηt δ σ 0.133 0.415 2.22 0.143 0.309 5 0.700 1.11 2.39 c/D n AR Y F N h d s tip t t 0.133 0.415 2.22 0.143 0.309 5 0.700 1.11 2.39 The first two parameters as well as N in Table 3 define the blade solidity from hub to tip. When calculating the solidity (c/s), it is found that it is below 0.7 along the entire radius The first two parameters as well as N in Table 3 define the blade solidity from hub of the blade even at the hub, as can be observed in Figure 8. Therefore, according to [46] to tip. When calculating the solidity (c/s), it is found that it is below 0.7 along the entire measurement data on isolated blade profiles—such as data included in Figure 4 in the radius of the blade even at the hub, as can be observed in Figure 8. Therefore, according case of a 8% cambered plate—can be applied in the present design of the PVS-affected to [46] measurement data on isolated blade profiles—such as data included in Figure 4 in rotor. In addition to the aerodynamic data in Figure 4, the empirical data on PVS in [5] the case of a 8% cambered plate—can be applied in the present design of the PVS-affected and [13] are also related to isolated profiles, thus fitting to the low-solidity approach uti- rotor. In addition to the aerodynamic data in Figure 4, the empirical data on PVS in [5,13] lized herein. are also related to isolated profiles, thus fitting to the low-solidity approach utilized herein. 0.75 0.60 0.45 0.30 0.15 0.00 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Figure 8. Blade solidity as a function of dimensionless radius (R = r/r ). tip Figure 8. Blade solidity as a function of dimensionless radius (R = r/rtip). In what follows, preliminary design efforts are made for obtaining a rotor blading exhibiting PVS of spanwise constant frequency in order to fulfill the pessimistic condi- In what follows, preliminary design efforts are made for obtaining a rotor blading tion outlined at the end of the Introduction. The PVS frequency can be expressed from exhibiting PVS of spanwise constant frequency in order to fulfill the pessimistic condition Equation (2) as follows: outlined at the end of the Introduction. The PVS frequency can be expressed from Equa- U (r ) tion (2) as follows: f (r ) = St = const (14) PVS b b(r ) U (r ) 0 b A nearly constant value off St* (r ) = 0.19 St was found = const for a 8% cambered plate for various (14) PVS b b(r ) Reynolds numbers and angles of attack in the present authors’ previous measurement campaign [13]. Keeping the uncertainty of the measurement-based St* data in mind, this A nearly constant value of St* ≈ 0.19 was found for a 8% cambered plate for various value is in fair agreement with the literature-based data in Equation (2). St* = 0.19 was used Reynolds numbers and angles of attack in the present authors’ previous measurement for the calculation presented in the paper. In order to clarify the trend of the free-stream campaign [13]. Keeping the uncertainty of the measurement-based St* data in mind, this velocity, the velocity vectors are shown in Figure 9. Based on Figure 9, the square of the value is in fair agreement with the literature-based data in Equation (2). St* = 0.19 was free-stream velocity can be written as follows [46]: used for the calculation presented in the paper. In order to clarify the trend of the free- stream velocity, the velocity vectors are shown  in Figure 9. Based on Figure 9, the square Dc (r ) 2 2 b of the free-stream velocity can be written as follows [46]: U (r ) = c (r ) + u(r ) (15) 0 x b b b Δc (r ) 2 2   u b U (r ) = c (r ) + u (r ) −   (15) where c is the axial velocity component; u is the circumferential velocity; and Dc is the x 0 b x b b u   increase of tangential velocity due to the rotor. where cx is the axial velocity component; u is the circumferential velocity; and Δcu is the increase of tangential velocity due to the rotor. c/s Int. J. Turbomach. Propuls. Power 2022, 7, 2 13 of 23 Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 13 of 23 Figure 9. Velocity triangles. Figure 9. Velocity triangles. From Equation (15) the square of the dimensionless free-stream velocity can be ex- From Equation (15) the square of the dimensionless free-stream velocity can be ex- pr pressed essed as as fo follows: llows: U (r )  ψ (R) U (r ) y (R) 2 t,is 2 0 b 0 b t,is 2 2 = ϕ (R) + R −  ≡ W (R) = j (R) + R  W (R ) (16) (16)   u 44RR tip tip   where W is the dimensionless free-stream velocity; j = c (r )/u is the local flow coefficient; where W is the dimensionless free-stream velocity; φ = cx(rb)/utip is the local flow coeffi- x b tip R is the dimensionless radius; and y is the local isentropic total pressure rise coefficient. cient; R is the dimensionless radius; and ψt,is is the local isentropic total pressure rise coef- t,is As ficient a brief . As appr a broximation, ief approxim the atcir ion, cumfer the ci ential rcumferent velocity ial v u,erlocit epresenting y u, repra esenting solid body a sor lid bod otation, y dominates in U . Therefore, U tends to approximately linearly increase with R. In a refined rotation, dominates in U0. Therefore, U0 tends to approximately linearly increase with R. 0 0 calculation, presented later, c /u and Dc /u are taken into account when obtaining U . In a refined calculation, presented x tip later, ucx/u tip tip and Δcu/utip are taken into account when 0 In order to provide a spanwise constant f , b(r ) tends to increase along the span by obtaining U0. PVS b such means that its increase matches with the spanwise increase of U . In order to provide a spanwise constant fPVS, b(rb) tends to increa 0se along the span by According to [5], the distance between vortex rows normalized by the blade chord is: such means that its increase matches with the spanwise increase of U0. According to [5], the distance between vortex rows normalized by the blade chord is: b(r ) Q(r ) b b = K (17) bc (r ) Θ c (r ) b b = K (17) c c where K*  1.2 is an empirical coefficient [5] and Q is the momentum thickness of the blade wake. According to [17,48] the drag coefficient can be written as: where K* ≈ 1.2 is an empirical coefficient [5] and Θ is the momentum thickness of the blade wake. According to [17,48] the drag coefficient can be written as: 2Q(r ) C (r ) = (18) D b 2Θ(r ) C (r ) = (18) D b Thus b can be expressed as follows: Thus b can be expressed as follows: C (r ) D b b(r ) = c K (19) C (r ) * D b b(r ) = c K (19) From Equation (19) it can be observed that a spanwise increase of b can be achieved via From Equation (19) it can be observed that a spanwise increase of b can be achieved spanwise increase of C . Based on Figure 4, C (R ) = 0.032 is chosen, which corresponds D D mid via spanwise increase of CD. Based on Figure 4, CD(Rmid) = 0.032 is chosen, which corre- to a = 6.8 . In the design of the PVS-affected rotor, the following range of C (R) was used: sponds to α = 6.8°. In the design of the PVS-affected rotor, the following range of CD(R) 0.0196  C  0.0410. Furthermore, as Figure 4 illustrates, the C data within the design D D was used: 0.0196 ≤ CD ≤ 0.0410. Furthermore, as Figure 4 illustrates, the CD data within the range are assigned to a data, and via such assignment, they also determine the design range design range are assigned to α data, and via such assignment, they also determine the for the local lift coefficient C (a). Therefore, the lift-to-drag ratio LDR(a) = C /C data are L L D design range for the local lift coefficient CL(α). Therefore, the lift-to-drag ratio LDR(α) = also obtained for the entire design range, incorporating the data at R . Thus, each of mid CL/CD data are also obtained for the entire design range, incorporating the data at Rmid. C (R ), C (R ), and LDR(R ) are available; these quantities will play an important D mid L mid mid Thus, each of CD(Rmid), CL(Rmid), and LDR(Rmid) are available; these quantities will play an role in the further investigation of the PVS-affected rotor, as presented later. important role in the further investigation of the PVS-affected rotor, as presented later. Iterative Method Iterative Method In order to provide a spanwise constant f , the nearly linear spanwise increase of PVS U is In order to matched in provide a sp blade design anwise const with the spanwise ant fPVS, the in ne crar ease ly line of C ar sp . T an o wise be able incrto ease design of U0 0 D such is ma atched fan, an in bla iterative de design wi method th the is elaborated spanwise i asnfollows, crease ofusing CD. To be the data ablein to desi Table gn su 3 as ch thea basis. fan, an Using iterative method is elabor Equation (16) as an ini ated tialas foll guess, ows, using the data the increase of tangential in Table velocity 3 as the basis due to. Using Equation (16) as an initial guess, the increase of tangential velocity due to the rotor Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 14 of 23 Int. J. Turbomach. Propuls. Power 2022, 7, 2 14 of 23 is neglected; therefore, W(R) is calculated from only the local flow coefficient φ = cx(rb)/utip and the dimensionless radius corresponding to the rigid body rotation. In the present the rotor is neglected; therefore, W(R) is calculated from only the local flow coefficient methodology, uniform axial inlet condition is assumed, cx(rb)/utip = constant. Furthermore, j = c (r )/u and the dimensionless radius corresponding to the rigid body rotation. In b tip based on [46] as an approximation, the change of meridional—i.e., axial—velocity is ne- the present methodology, uniform axial inlet condition is assumed, c (r )/u = constant. x tip glected through the rotor. The above implies that spanwise constant axial velocity is pre- Furthermore, based on [46] as an approximation, the change of meridional—i.e., axial— sumed, as a brief approximation. velocity is neglected through the rotor. The above implies that spanwise constant axial velocity As a ne is prxt s esumed, tep, ba as sea d on the val brief approximation. ue of W(Rmid)/CD(Rmid), the drag coefficients are deter- mined Asalon a next g the blade step, based span. As desc on the value ribed earlier, the of W(R )/CCD (R (R), α ),(R the ) and drag CLcoef (R) d ficients ata are as are- mid D mid determined along the blade span. As described earlier, the C (R), a(R) and C (R) data are signed to each other, via Figure 4. As a next stage of the design, Δcu/utip is expressed from D L assigned the simpli to fied work equa each other, via tion of Figur a en4 el . As ementa a next l rotor: stage of the design, Dc /u is expressed tip from the simplified work equation of an elemental rotor: c 2Δc (r ) u b C (r ) ≅ (20) L b c s(r ) 2DUc ( (rr )) b 0 bb C (r ) (20) L b s(r ) U (r ) b 0 b where s = 2rbπ/N is the blade spacing. With the knowledge of Δcu(R)/utip, the local isen- tropic total pressure rise coefficient ψt,is(R) is expressed from the Euler equation of tur- where s = 2r /N is the blade spacing. With the knowledge of Dc (R)/u , the local b u tip bomachines: isentropic total pressure rise coefficient y (R) is expressed from the Euler equation of t,is turbomachines: Δp (r ) = ρ u (r )Δc (r ) t,is b a b u b (21) Dp (r ) = r u(r )Dc (r ) (21) a u t,is b b b here Δpt,is is the isentropic total pressure rise, and ρa is the density of air. This computed here Dp is the isentropic total pressure rise, and r is the density of air. This computed t,is ψt,is(R) is then substituted into Equation (16) for calculating a new approximation of W(R) y (R) is then substituted into Equation (16) for calculating a new approximation of W(R) t,is in the next iteration loop. Fast convergence is obtained in two to three iteration loops. The 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 ψt,is(R) for the consecutive results are judged to be converged if the relative difference in y (R) for the consecutive t,is iteration steps becomes less than 2%. iteration steps becomes less than 2%. A feature of the iterative method is that ψt,is(R) and α(R) are actually the results of the A feature of the iterative method is that y (R) and a(R) are actually the results of t,is design process. The spanwise distributions of the calculated quantities are shown in Fig- the design process. The spanwise distributions of the calculated quantities are shown ure 10, where γ(R) is stagger angle measured from the circumferential direction. Annulus- in Figure 10, where g(R) is stagger angle measured from the circumferential direction. averaging of ψt,is(R) represents the global isentropic total pressure rise obtained as Ψt/ηt, Annulus-averaging of y (R) represents the global isentropic total pressure rise obtained t,is using the data in Table 3. as Y /h , using the data in Table 3. t t Figure 10. Calculated distributions y (R), a(R), and g(R) as a function of R. t,is Figure 10. Calculated distributions ψt,is(R), α(R), and γ(R) as a function of R. Figure 11 represents the proportionate three-dimensional model of the designed Figure 11 represents the proportionate three-dimensional model of the designed PVS-affected fan. The rotor has a realistic, common geometry, as it is representative of pr PVS-affected opeller-type fan axial . The flow roto industrial r has a refans. alistic, The common blade geometry, geometry isas it in accor is represen dance with tative the of propeller-type axial flow industrial fans. The blade geometry is in accordance with the fact that, as a manufacturing simplification, low-speed axial fan blades of cost-effective manufacturing fact that, as a manu can be fact made uring simpl from a plate ificati of on, low spanwise -speed ax constant ial fchor an blade d, as s they of cos aret-effe rolled ctive in such manu afact way urthat ing can be m spanwise ad constant e from a p camber late of spanw and moderate ise const twist ant chor occur d, a . s they are rolled in such a way that spanwise constant camber and moderate twist occur. Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 15 of 23 Int. J. Turbomach. Propuls. Power 2022, 7, 2 15 of 23 Figure 11. The PVS-affected fan rotor. Figure 11. The PVS-affected fan rotor. 4.2. Mechanically or Acoustically Unfavorable Design Cases 4.2. Mechanically or Acoustically Unfavorable Design Cases In order to set up mechanically unfavorable design cases, representing a coincidence In order to set up mechanically unfavorable design cases, representing a coincidence of the frequency of PVS with the frequency of the first bending mode of blade vibration, it of the frequency of PVS with the frequency of the first bending mode of blade vibration, is necessary to express U and b in Equation (14). However, b was previously expressed, it is necessary to express U0 and b in Equation (14). However, b was previously expressed, as shown in Equation (17). Therefore, it is only necessary to deal with U . Based on as shown in Equation (17). Therefore, it is only necessary to deal with U0. Based on Equa- Equations (20) and (21), the free-stream velocity is written as follows: tions (20) and (21), the free-stream velocity is written as follows: 2Dp (r ) s(r ) t,is 2Δp (b r ) s(rb) t,is b b U (r ) = (22) 0 b U (r ) = (22) 0 b r u(r ) c C (r ) a b L b ρ u(r ) c C (r ) a b L b Thus, the PVS frequency can be expressed from Equations (14), (19) and (22): Thus, the PVS frequency can be expressed from Equations (14), (19) and (22): 2D2pΔp(r(r)) s(r ) 2 t,is s(r ) 2 b b * t,is b f (r f ) (= r )S=t St (23) PVS b (23) PVS b r c C (r ) u(r ) K C (r ) c a ρ cL C (br ) u(r b ) K C D(r ) bc a L b b D b With the use of s(r )/u(r ) = (2 r /N)/(2 r n) = 1/(nN) and C = C /LDR, Equation (23) With the use of sb (rb)/u(brb) = (2 r b bπ/N)/(2 rb bπn) = 1/(nN) and CD D = CL L/LDR, Equation (23) is written as follows: is written as follows: 2Dp (r ) 1 2 LDR(r ) 2Δp (r ) t,is b 1 2 LDR(r ) b * t,is b f (r ) = St (24) PVS f (r ) = St (24) PVS b r c C (r ) N n K C (r ) c a L L ρ c C (b r ) N n K C (r b ) c a L b L b where N is the blade count and n is the rotor speed. The right-hand side of Equation (24) where N is the blade count and n is the rotor speed. The right-hand side of Equation (24) can be rearranged according to Equation (25) in the following way: In the first term, the can be rearranged according to Equation (25) in the following way: In the first term, the global aerodynamic performance characteristics are grouped. The second term represents global aerodynamic performance characteristics are grouped. The second term represents the operational condition of the rotor. In the third term, the basic parameters of the blade the operational condition of the rotor. In the third term, the basic parameters of the blade geometry are gathered. The fourth term represents the aerodynamics parameters of the geometry are gathered. The fourth term represents the aerodynamics parameters of the elemental blade section. Finally, the last term incorporates the empirics and coefficients. elemental blade section. Finally, the last term incorporates the empirics and coefficients. Dp (r ) 1 1 LDR(r ) 4 St Δp (r ) 1 1 LDR(r ) 4 St t,is b b t,is b f (fr ) (= r ) = (25) PVS b (25) PVS b 2 2 2 2 * r n K N c C (r ) aρ n N c C (r ) K a L L b In In order order t to o obt obtain ain gener generalized alized conc conclusions lusions a and nd tr trends, di ends, dimensionless mensionless q quantities uantities and and gr groups m oups must ust be be introduced. It should be introduced. It should be emphasized emphasized that PVS that PVS fr frequency is constant along equency is constant along the blade span due to the presented fan design method. Therefore, any characteristic taken the blade span due to the presented fan design method. Therefore, any characteristic taken at an arbitrary r radius can be replaced by the same characteristic taken at R . Further- at an arbitrary rb radius can be replaced by the same characteristic taken at Rmid. Further- b mid more, as a brief approximation h (r ) = h (R )  constant is presumed. With respect to more, as a brief approximation ηtt(rb) = ηt(tRmid) ≈ constant is presumed. With respect to the b mid the foregoing, the first term on the right-hand side of Equation (25) can be expressed: foregoing, the first term on the right-hand side of Equation (25) can be expressed: Δp (r ) Δp (R ) Δp (R ) Ψ (R ) t,is b t,is mid tip Dp (r ) Dp (R ) t mid t mid Dp (R ) Y (R ) tip t,is t,is t t b mid mid mid = = = (26) = = = (26) ρ ρ ρ η (R ) η 2 r a r a r h a ( tR mid ) t h 2 a a a t mid t where utip = Dtipπn is the tip circumferential speed. The second and the third terms are written as follows: Int. J. Turbomach. Propuls. Power 2022, 7, 2 16 of 23 where u = D n is the tip circumferential speed. The second and the third terms are tip tip written as follows: 1 1 1 1 1 1 1 = D  = s (27) tip tip 2 2 2 n c N n D  c N u c tip tip Thus Equation (25): s u s Y (R ) 1 LDR(R ) 4St Y (R ) LDR(R ) 2 St tip tip tip tip t mid mid t mid mid f (R ) = = (28) PVS mid 2 2  2 2 h 2 u c C (R ) K h (R ) c C (R ) K t tip t mid mid mid L L As it is outlined previously, the mechanically unfavorable design case is considered when the spanwise constant PVS frequency coincides with the first bending eigenfrequency of the fan blade, posing an increased risk of blade resonance: f = const = f (29) PVS B,1 If Equations (28) and (13) are substituted in the right-hand side and the left-hand side of Equation (29), respectively, the following equation is obtained: u s Y (R ) LDR(R ) 2 St 0.56 t tip tip mid mid = K t a (30) b b 2 2  2 h (R ) c C (R ) K l mid mid After rearrangement, Equation (30) can be written in the following as dimensionless, from: u C (R ) h (R ) t c 1 0.56 K tip t mid mid = K (31) a Y (R ) c s LDR(R ) 2St tip AR b mid mid Equation (31) provides a means for calculating the critical u /a velocity ratio for tip b 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 can be obtained (cf. Equation (10) and the paragraph below), and thus, the critical u value can be computed. Hence, critical tip rotor diameter X rotor speed data couples can be discovered for an entire rotor family, consisting of rotors of various diameters and speeds. If the rotor diameter and the nominal rotor speed are fixed for further defining a specific case study, the resultant, nominal u value can be compared to the aforementioned tip 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 , using Equation (25), for acoustics evaluation. PVS 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 away from this plateau in blade design, by selecting the appropriate operational and PVS geometrical characteristics, gives a potential for moderating the impact of fan noise on humans. If such design intent cannot be realized for modifying f , the PVS phenomenon PVS 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. Int. J. Turbomach. Propuls. Power 2022, 7, 2 17 of 23 5. Calculation Example for the Designed Rotor First, based on Equation (31) and data in Table 3, as it is specific to the PVS-affected rotor designed herein, the critical tip speeds are computed for various blade materials, which means different values of a in terms of the calculation process. The calculated values are summarized in Table 4. Table 4. Critical rotor tip speeds. Steel Aluminum Polycarbonate a [m/s] 5000 5100 1350 u [m/s] 1.80 1.83 0.49 tip,crit The methodology presented herein provides a means for simply checking whether PVS may cause a risk at all from a blade resonance point of view. Based on Table 4, it can be concluded that in the present case study, the critical tip speed is sufficiently low to make these cases irrelevant for the blade resonance point of view of the first-order bending mode (Figure 6). Namely, only impractically low rotor diameters D and/or rotor speeds n would result in coincidence of the f and f [Equation (10)] values for the presented PVS B,1 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. The reader is reminded that the mechanical effect of the blade twist is neglected herein, cf. Section 2. With consideration of the blade twist, the first bending eigenfrequency would be less, thus reducing the impractically low critical tip speed even further. As the second part of the calculation example, the rotor diameter and rotor speed are fixed as follows: D = 0.900 m, n = 1450 1/min. Such values are relevant in industrial tip ventilation. They result in u = 68.3 m/s. Considering data in Table 4 as well as previously tip 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. Table 5. PVS frequency for a five-bladed, n = 1450 1/min axial flow fan. D c C (R ) LDR (R ) s f tip L mid mid tip PVS 0.900 m 0.120 m 1.30 43.5 0.565 4520 Hz The calculated PVS frequency falls within the third-octave band of the middle fre- quency of 5 kHz. Thus, it approximates the plateau of the A-weighting graph. Therefore, the related noise may cause increased annoyance for a human observer. However, by mod- ifying the design parameters, the axial fan can be redesigned to keep the PVS frequency away from the plateau of the A-weighting graph, based on the presented computation. The systematic exploration of the advantageous parameter modifications, while finding reasonable compromises with other design perspectives, is also a future task. Based on Table 2, for the first bending eigenfrequency of the designed blade, the following value is calculated: f = 116 Hz. For the sake of completeness, the following B,1 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 fre- quency. The rotational speed n = 1450 1/min, which corresponds to 24 Hz, is thus  20% of the first bending eigenfrequency, which is satisfactory. On the other hand, as mentioned in Section 3.1, elements located upstream of the rotor may lead to rotor–stator interaction, occurring at a frequency of rotational frequency n Int. J. Turbomach. Propuls. Power 2022, 7, 2 18 of 23 multiplied by the number of upstream elements. Therefore, it is necessary to examine for which number of upstream elements (e.g., support strut, inlet guide vane) the computed eigenfrequency would coincide with the excitation frequency of the rotor–stator interaction. As f  5  24 Hz = 120 Hz, the critical element number is five, which should be avoided B,1 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 Based on the semi-empirical model in literature, the pessimistic design condition of spanwise constant PVS frequency is determined from an aerodynamic approach. To fulfill this condition, an iterative fan design method was elaborated, resulting in a design case study of a PVS-affected rotor. The frequency of PVS was computed with the knowledge of the global operational and geometrical characteristics of an axial fan. A calculation process for determining the eigenfrequency related to the first bending mode of vibration of a circular-arc-cambered plate blade was presented. By combining these two approaches, guidelines can be formulated, already in the preliminary design phase, for the following. (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. As a future objective, opposing the pessimistic design scenarios, redesign efforts are to be made for moderating/avoiding blade resonance and/or noise annoyance. Such efforts incorporate actions against the constancy of PVS frequency along the span. Systematic redesign scenarios as well as comparative experiments—also incorporating pessimistic, PVS-affected rotor cases—will be realized in the future for validating the methodology presented herein. The design aspects related to PVS are to be investigated in the future via studies on 2D—i.e., rectilinear—blade models as well as on truly 3D rotor blade geometries— unavoidable for full consideration of realistic rotor flow effects—by the concerted means of computational fluid dynamics (CFD), analytical mechanics, finite-element mechanical computations, and experimentation. Such studies will also serve as the exploration of the effects of assumptions and simplifications made in the method presented herein. Author Contributions: G.D. and J.V. conceptualization, methodology, investigation, and writing. All authors have read and agreed to the published version of the manuscript. Funding: This work has been supported by the Hungarian National Research, Development and Innovation (NRDI) Centre under contract No. NKFI K 129023. The research reported in this paper and carried out at BME has been supported by the NRDI Fund (TKP2021, project no. BME-NVA-02, TKP2021-EGA, and TKP2020 NC, Grant No. BME-NCS) based on the charter of bolster issued by the NRDI Office under the auspices of the Ministry for Innovation and Technology of Hungary. The contribution of Gábor DAKU has been supported by Gedeon Richter Talent Foundation (registered office: 1103 Budapest, Gyömroi ˝ út 19-21.), established by Gedeon Richter Plc., within the framework of the “Gedeon Richter PhD Scholarship”. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Acknowledgments: The authors are thankful to Tamás KALMÁR-NAGY for his assistance in the blade mechanics analysis. Conflicts of Interest: The authors declare no conflict of interest. Int. J. Turbomach. Propuls. Power 2022, 7, 2 19 of 23 Abbreviations The following abbreviations are used in this manuscript: Latin Letters A Cross-section [m ] 1/2 a Wave propagation speed = (E/r ) [m/s] b b AR Aspect ratio = l/c [-] b Transversal dist. between vortex rows [m] c/s Blade solidity [-] D Rotor diameter [m] d Trailing edge thickness [m] TE 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 ] I Polar moment of area [m ] I Torsional constant [m ] K* Empirical coefficient  1.2 [-] K Preliminary design constant [-] K Blade mechanics coefficient [-] K Mechanical constant [-] K Bending mode constant [-] B,i c Blade chord length [m] c Axial velocity component [m/s] C Drag coefficient [-] C Lift coefficient [-] l Blade span [m] LDR Lift-to-drag ratio = C /C [-] L D N Blade count [-] n Rotor speed [1/s] R Dimensionless radius = r/r [-] tip r Radial coordinate [m] Re Reynolds number = c U /n [-] c a s Blade spacing = 2r/N [m] St Strouhal number [-] St* Universal Strouhal number [-] t Profile thickness [m] t/c Relative thickness [-] u Rotor circumferential velocity [m/s] U Free-stream velocity = (w + w )/2 [m/s] 0 1 2 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] Int. J. Turbomach. Propuls. Power 2022, 7, 2 20 of 23 Greek Letters a Angle of attack [ ] b l Vibration constant  (2i 1)/2 g Stagger angle [ ] d Specific diameter [-] z Rotation angle [rad] Dc Increase of tangential velocity [m/s] Dp Isentropic total pressure rise [Pa] t,is h Efficiency [-] Q Momentum thickness of blade wake [m] n Hub-to-tip ratio [-] n Kinematic viscosity of air [m /s] n Poisson’s ratio [-] r Density [kg/m ] s Specific speed [-] t Time [s] j Local flow coefficient [-] F Global flow coefficient [-] y Local pressure rise coefficient [-] Y 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 Int. J. Turbomach. Propuls. Power 2022, 7, 2 21 of 23 Appendix A Table A1. Third-octave band. Lower Band Limit Center Frequency Upper Band Limit Band No. (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 References 1. Brooks, T.F.; Pope, D.S.; Marcolini, M.A. Airfoil Self-Noise and Prediction; NASA Ref. Publication: Washington, DC, USA, 1989; Volume 1218. Available online: https://ntrs.nasa.gov/citations/19890016302 (accessed on 27 September 2021). 2. 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Available online: http://hdl.handle.net/10890/13530 (accessed on 27 September 2021). 6. Tam, C.K.W. Discrete tones of isolated airfoils. J. Acoust. Soc. Am. 1974, 55, 1173–1177. [CrossRef] 7. Wright, S. The acoustic spectrum of axial flow machines. J. Sound Vib. 1976, 45, 165–223. [CrossRef] 8. Nash, E.C.; Lowson, M.V.; McAlpine, A. Boundary-layer instability noise on aerofoils. J. Fluid Mech. 1999, 382, 27–61. [CrossRef] 9. Ausoni, P.; Farhat, M.; Bouziad, Y.; Kueny, J.-L.; Avellan, F. Kármán vortex shedding in the wake of a 2D hydrofoil: Measurement and numerical simulation. In Proceedings of the IAHR International Meeting of WG on Cavitation and Dynamic Problems in Hydraulic Machinery and Systems, Barcelona, Spain, 28–30 June 2006; p. 14. Available online: https://infoscience.epfl.ch/ record/88122 (accessed on 27 September 2021). Int. J. Turbomach. Propuls. Power 2022, 7, 2 22 of 23 10. Roger, M.; Moreau, S. Extensions and limitations of analytical airfoil broadband noise models. 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In Proceedings of the European Conference on Turbomachinery Fluid Dynamics and hermodynamics, Stockholm, Sweden, 3–7 April 2017; European Turbomachinery Society: Lausanne, Switzerland, 2017; p. 13. 23. Masi, M.; Lazzaretto, A. A New Practical Approach to the Design of Industrial Axial Fans: Tube-Axial Fans with Very Low Hub-to-Tip Ratio. J. Eng. Gas Turbines Power 2019, 141, 101003. [CrossRef] 24. Carnegie, W. Vibrations of Pre-Twisted Cantilever Blading. Proc. Inst. Mech. Eng. 1959, 173, 343–374. [CrossRef] 25. Gruber, J.; Blahó, M.; Herzog, P.; Kurutz, I.; Preszler, L.; Vajna, Z.; Szentmártony, T. Fans (Ventilátorok, in Hungarian), 2nd ed.; Muszaki ˝ Könyvkiadó: Budapest, Hungary, 1968. 26. Bohl, W. Strömungsmaschinen 2; Berechnung und Konstruktion; Kamprath-Reihe: Vogel Buchverlag, Germany, 1991. 27. Banerjee, J.; Williams, F. Coupled bending-torsional dynamic stiffness matrix for timoshenko beam elements. Comput. Struct. 1992, 42, 301–310. [CrossRef] 28. Sinha, S.K. Combined Torsional-Bending-Axial Dynamics of a Twisted Rotating Cantilever Timoshenko Beam with Contact-Impact Loads at the Free End. J. Appl. Mech. 2006, 74, 505–522. [CrossRef] 29. Sinha, S.K.; Turner, K.E. Natural frequencies of a pre-twisted blade in a centrifugal force field. J. Sound Vib. 2011, 330, 2655–2681. [CrossRef] 30. Sun, Q.; Ma, H.; Zhu, Y.; Han, Q.; Wen, B. Comparison of rubbing induced vibration responses using varying-thickness-twisted shell and solid-element blade models. Mech. Syst. Signal Process. 2018, 108, 1–20. [CrossRef] 31. Rao, S.S. Vibration of Continuous Systems; John Wiley & Sons: Hoboken, NJ, USA, 2007. 32. Gorman, D.J. Free Vibration Analysis of Beams and Shafts; John Wiley: New York, NY, USA, 1975. 33. Bishop, R.; Price, W. Coupled bending and twisting of a timoshenko beam. J. Sound Vib. 1977, 50, 469–477. [CrossRef] 34. Bercin, A.; Tanaka, M. Coupled Flexural–Torsional Vibrations of Timoshenko Beams. J. Sound Vib. 1997, 207, 47–59. [CrossRef] 35. 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Loya, J.; Rubio, M.L.; Fernandez-Saez, J. Natural frequencies for bending vibrations of Timoshenko cracked beams. J. Sound Vib. 2006, 290, 640–653. [CrossRef] 43. Balla, E.; Vad, J. Lift and drag force measurements on basic models of low-speed axial fan blade sections. Proc. Inst. Mech. Eng. Part A J. Power Energy 2018, 233, 165–175. [CrossRef] 44. Wallis, R.A. Axial Flow Fans; George Newnes Ltd.: London, UK, 1961. 45. Beer, F.P.; Johnston, E.R.; Mazurek, D.F. Vector Mechanics for Engineers: Statics, 11th ed.; McGraw-Hill: New York, NY, USA, 2016; pp. 485–572. 46. Carolus, T. Ventilatoren; B. G. Teubner Verlag: Wiesbaden, German, 2003. Int. J. Turbomach. Propuls. Power 2022, 7, 2 23 of 23 47. Regulation of the European Commission (EU). No. 327/2011, of 30 March 2011, implementing Directive 2009/125/EC of the European Parliament and of the Council of the European Union with regard to ecodesign requirements for fans driven by motors with an electric input power between 125 W and 500 kW. Off. J. Eur. Union 2011, 54, 8–21. 48. Fathy, A.; Rashed, M.; Lumsdaine, E. A theoretical investigation of laminar wakes behind airfoils and the resulting noise pattern. J. Sound Vib. 1977, 50, 133–144. [CrossRef] 49. Norton, M.P.; Karczub, D.G. Fundamentals of Noise and Vibration Analysis for Engineers, 2nd Edition. Noise Control. Eng. J. 2007, 55, 275. [CrossRef] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png International Journal of Turbomachinery, Propulsion and Power Multidisciplinary Digital Publishing Institute

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

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International Journal of Turbomachinery Propulsion and Power Article Preliminary Design Guidelines for Considering the Vibration and Noise of Low-Speed Axial Fans Due to Profile Vortex Shedding Gábor Daku * and János Vad Department of Fluid Mechanics, Faculty of Mechanical Engineering, Budapest University of Technology and Economics, Bertalan Lajos u. 4-6, H-1111 Budapest, Hungary; vad@ara.bme.hu * Correspondence: daku@ara.bme.hu † This paper is a revised version of our paper published in Proceedings of the European Turbomachinery Conference ETC14, Gdansk, Poland, 12–16 April 2021. Abstract: This paper presents a critical overview on worst-case design scenarios for which low-speed axial flow fans may exhibit an increased risk of blade resonance due to profile vortex shedding. To set up a design example, a circular-arc-cambered plate of 8% relative curvature is investigated in twofold approaches of blade mechanics and aerodynamics. For these purposes, the frequency of the first bending mode of a plate of arbitrary circular camber is expressed by modeling the fan blade as a cantilever beam. Furthermore, an iterative blade design method is developed for checking the risky scenarios for which spanwise and spatially coherent shed vortices, stimulating pronounced vibration and noise, may occur. Coupling these two approaches, cases for vortex-induced blade resonance are set up. Opposing this basis, design guidelines are elaborated upon for avoiding such resonance. Based on the approach presented herein, guidelines are also developed for moderating the annoyance due to the vortex shedding noise. Keywords: axial flow fan; blade vibration; low tip speed; preliminary fan design; vortex shedding Citation: Daku, G.; Vad, J. Preliminary Design Guidelines for Considering the Vibration and Noise 1. Introduction of Low-Speed Axial Fans Due to Vortex shedding (VS) from low-speed axial flow fan rotor blades has become of Profile Vortex Shedding. Int. J. Turbomach. Propuls. Power 2022, 7, 2. engineering relevance in the past decades. The VS phenomenon discussed in this paper— https://doi.org/10.3390/ijtpp7010002 termed herein as profile vortex shedding (PVS), and illustrated in Figure 1—is not to be confused with the trailing-edge-bluntness, VS, which takes place past the blunt trailing edge Received: 28 September 2021 (TE) of the blade profile, acting as the aft portion of a bluff body [1]. In the aforementioned Accepted: 4 January 2022 literature, PVS is referred to as a laminar-boundary-layer VS, because it can only occur Published: 7 January 2022 if the boundary layer is initially laminar at least over one side of the blade profile. In Publisher’s Note: MDPI stays neutral this case, the initially laminar boundary layer, being separated near or after mid-chord with regard to jurisdictional claims in position, reattaches in the vicinity of the TE—thus resulting in a separation bubble—and published maps and institutional affil- finally undergoes a laminar-to-turbulent transition. Moreover, the position and the size iations. of the formed separation bubble plays a key role in tonal PVS noise emission. Recently, Yakhina at al. [2] published a detailed investigation about tonal TE noise radiated by low Reynolds number airfoils. They observed that a precondition for tonal noise emission is the formed separation bubble being sufficiently close to the TE. PVS may occur within a Copyright: © 2022 by the authors. certain Reynolds number range. Based on the literature [3–5], a lower limit of Re = 5  10 Licensee MDPI, Basel, Switzerland. is assumed herein, while the upper limit is determined by the critical Reynolds number This article is an open access article of the natural laminar-to-turbulent transition. When PVS is discussed for low-speed fans, distributed under the terms and as in the present paper, incompressible flow is considered by implying a Mach number conditions of the Creative Commons of 0.3. Attribution (CC BY-NC-ND) license (https://creativecommons.org/ licenses/by-nc-nd/4.0/). Int. J. Turbomach. Propuls. Power 2022, 7, 2. https://doi.org/10.3390/ijtpp7010002 https://www.mdpi.com/journal/ijtpp Int. J. Turbomach. Propuls. Power 2022, 7, 2 2 of 23 Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 2 of 23 Figure 1. Profile vortex shedding (PVS). Figure 1. Profile vortex shedding (PVS). V V arious arious model modelssar are e available available in in the li the literatur terature on the PVS phenomenon. In th e on the PVS phenomenon. In this is paper, paper, only only the c the classic lassic model by model by Tam Tam [6] [6] and and Wr Wright ight [[7] is re 7] is referr ferred ed to, in to, in orde orderr to to pr provide ovide aa straightforward and comprehensive interpretation on the mechanism. According to this straightforward and comprehensive interpretation on the mechanism. According to this model, PVS is related to a self-excited feedback loop. Due to the unstable laminar boundary model, PVS is related to a self-excited feedback loop. Due to the unstable laminar bound- layer, Tollmien–Schlichting instability waves are generated; these waves travel downstream ary layer, Tollmien–Schlichting instability waves are generated; these waves travel down- toward the TE where sound scattering occurs, and acoustic waves are created. The acoustic stream toward the TE where sound scattering occurs, and acoustic waves are created. The waves propagate upstream to amplify the original instabilities. If appropriate phase acoustic waves propagate upstream to amplify the original instabilities. If appropriate conditions are fulfilled, the disturbances are amplified at some frequencies, thus closing the phase conditions are fulfilled, the disturbances are amplified at some frequencies, thus feedback loop. Later, a number of authors, e.g., Nash et al. [8], carried out a critical revision closing the feedback loop. Later, a number of authors, e.g., Nash et al. [8], carried out a on the aforementioned feedback loop model. PVS may generate vibration on the blade. critical revision on the aforementioned feedback loop model. PVS may generate vibration As the studies by Ausoni et al. [9] suggest, the mechanisms of periodic vortex shedding on the blade. As the studies by Ausoni et al. [9] suggest, the mechanisms of periodic vortex and periodic blade vibration may mutually be coupled at a blade eigenfrequency, within a shedding and periodic blade vibration may mutually be coupled at a blade eigenfre- “lock-in” phenomenon. quency, within a “lock-in” phenomenon. In the case of low-speed axial flow fan blades, the difference between PVS and TE- In the case of low-speed axial flow fan blades, the difference between PVS and TE- bluntness VS in their physical mechanisms manifests itself in scaling techniques and the bluntness VS in their physical mechanisms manifests itself in scaling techniques and the values of the Strouhal number being also different. values of the Strouhal number being also different. For blade profiles with thick or blunt trailing edges, the TE-bluntness vortex shed- For blade profiles with thick or blunt trailing edges, the TE-bluntness vortex shed- ding [10] can be characterized by the Strouhal number based on the free-stream velocity U ding [10] can be characterized by the Strouhal number based on the free-stream velocity and the TE thickness d : TE U0 and the TE thickness dTE: St = f d /U = 0.20 (1) TE TE TE St = f d / U ≅ 0.20 TE TE TE 0 (1) For PVS [11,12], which is associated with the boundary layer transition and feedback mechanism: For PVS [11,12], which is associated with the boundary layer transition and feedback St = f b/U = 0.18 (2) PVS 0 mechanism: where f is the dominant frequency of the two types of VS, and b is the distance between St = f b / U ≅ 0.18 (2) PVS 0 the vortex rows. Yarusevych et al. [11] found that St* is universally valid for symmetrical, where f is the dominant frequency of the two types of VS, and b is the distance between relatively thick NACA airfoils for certain ranges of the Reynolds number and angle of attack, the vortex rows. Yar suggesting the usevy appellation ch et al. [11] of afound that “universal” St* Str is un ouhal iver number sally valSt id fo *, as r sym specified metrica in l, Equation relatively(2). thiHowever ck NACA , these airfoi airfoils ls for cert area not in rwidely anges of used the Reyno in axial l fan ds number and application; ther angle eforo e,f the present authors have extended the proposed St* definition as follows. Systematic attack, suggesting the appellation of a “universal” Strouhal number St*, as specified in wind-tunnel Equation (2). experiments However, th wer ese a e performed irfoils are not on widely blade section used in models axial ftypical an applfor icat low-speed ion; there- axial fans, using a single-component hot-wire probe. Based on the measured f and b values, fore, the present authors have extended the proposed St* definition as follows. Systematic the authors also confirmed the validity of St* for asymmetrical profile geometries, such as wind-tunnel experiments were performed on blade section models typical for low-speed 8% cambered plate and RAF-6E profiles, in a quasi-2D experimental analysis [13]. Thus, axial fans, using a single-component hot-wire probe. Based on the measured f and b val- the available experimental database on PVS frequency [5] was extended. ues, the authors also confirmed the validity of St* for asymmetrical profile geometries, From an engineering point of view, the practical aspects of PVS are remarkable in two such as 8% cambered plate and RAF-6E profiles, in a quasi-2D experimental analysis [13]. ways: vibration and noise. On the one hand, PVS creates a periodically fluctuating force Thus, the available experimental database on PVS frequency [5] was extended. normal to the chord, increasing the risk of blade vibration. On the other hand, VS appears From an engineering point of view, the practical aspects of PVS are remarkable in as the primary source of the aeroacoustics noise of low-speed axial flow fans [14–16]. Hence, two ways: vibration and noise. On the one hand, PVS creates a periodically fluctuating it is an important engineering objective to check and possibly to control the vibration and force normal to the chord, increasing the risk of blade vibration. On the other hand, VS noise of low-speed axial fans due to the PVS that is already in the preliminary design appears as the primary source of the aeroacoustics noise of low-speed axial flow fans [14– phase. It is worth noting that the signatures of blade vibration and PVS-related noise may 16]. Hence, it is an important engineering objective to check and possibly to control the coincide in the frequency spectra. In [17], the vibrometer connected to the airfoil indicated vibration and noise of low-speed axial fans due to the PVS that is already in the prelimi- a vortex-induced vibration, the dominant frequency of which coincided with that of the nary design phase. It is worth noting that the signatures of blade vibration and PVS-re- far-field tone. lated noise may coincide in the frequency spectra. In [17], the vibrometer connected to the Measurements on PVS in the literature are mostly related to isolated and steady airfoil indicated a vortex-induced vibration, the dominant frequency of which coincided airfoils [2,11–13,17,18]. Only a few studies dealt with PVS in the case of rotating blades of with that of the far-field tone. asymmetrical profiles being characteristic for realistic axial fans. Longhouse [19] detected Int. J. Turbomach. Propuls. Power 2022, 7, 2 3 of 23 PVS noise on an axial fan of four cambered plate blades with a constant blade chord. Nevertheless, except for a small segment (~10%) of the span of one blade in near-tip region, the PVS noise was suppressed by aft-chord serrations. Furthermore, the spanwise variation of free-stream velocity tended to broaden the noise signature of PVS, thus acting against a remarkable, well-detectable tonal PVS character. Grosche and Stiewitt [20] examined a four-bladed propeller-type axial fan rotor with a moderate sweep and twist. They observed PVS noise at a  4 angle of attack, viewed in the rotating frame of reference near the blade tip, for three different Reynolds numbers based on the chord length (Re = 9  10 , 5 5 1.3  10 , 2.6  10 ). 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]. In order to moderate the harmful noise and vibration effects due to PVS, pessimistic design scenarios have systematically been discovered for which such harmful effects are pronounced. Furthermore, these design characteristics were coupled with rotor dynamics consideration. Based on this coupling, an exemplary unfavorable design case was set up in terms of both operational and geometrical characteristics, for which PVS demonstrates an increased risk of blade resonance. Taking the rotor of such an unfavorable design case—termed hereafter the PVS-affected rotor—as reference, approximate semi-empirical design guidelines can already be elaborated in the preliminary design phase for checking and possibly avoiding PVS-induced resonance. The design guidelines for noise reduction, reported in [13], have also been further developed toward a more detailed model, as illustrated in this paper. 2. Blade Vibration: An Overview Turbomachinery blade profiles are suggested to be modeled in preliminary analysis as simple cantilever beams [24–30], which means that all degrees of freedom of the blades at the blade root, i.e., where connected to the hub, are constrained. The pure bending— or, in other words, transversal or flexural—vibrations of a prismatic beam with uniform cross-section according to time (t) and the coordinate along the longitudinal direction of the beam (z) are described by the partial differential equation below, which is derived from the Euler-Bernoulli’s beam theory [26,31]: 4 2 ¶ g(z, t) ¶ g(z, t) EI + r A = 0 (3) 4 2 ¶x ¶t where g(z, t) is the lateral displacement along the axis perpendicular to the blade chord; E is the Young modulus; I is the second moment of area with respect to the axis being parallel to the chord and fitting to the center of gravity (CG) of the blade section; A is the cross-sectional area of the blade profile; and r 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: (cos b l + cosh b l) i i F (z) = (cos b z cosh b z) (sin b z sinhb z) (4) i i i i i (sin b l sinhb l) i i where i is the order-number, i.e., 1, 2, 3 etc.; F(z) is the characteristic function or the normal mode of the beam; l is the radial extension of blade from hub to tip, i.e., the blade span, and b l  (2i 1)/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. Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 4 of 23 for some bending modes, generated on the basis of Equation (4). The vertical axis repre- Int. J. Turbomach. Propuls. Power 2022 sents the normal mode , 7, 2 (i.e., dimensionless displacement) and the horizontal axis shows 4 of 23 the dimensionless length of the beam. Figure 2. Normal bending modes of the beam. Figure 2. Normal bending modes of the beam. The pure torsion of the cantilever beam is governed by the following differential The pure torsion of the cantilever beam is governed by the following differential Equation [32]: 2 2 Equation [32]: ¶ z(z, t) ¶ z(z, t) GI + r I = 0 (5) t b p 2 2 2 2 ¶x ¶t ∂ ζ() z,τ ∂ ζ() z,τ GI + ρ I = 0 (5) t b p 2 2 where z(z, t) is the rotation angle around the longitudinal direction of the beam; G is the ∂x ∂τ shear modulus; I is the torsional stiffness or torsional constant; and I is the polar moment t p where ζ(z, τ) is the rotation angle around the longitudinal direction of the beam; G is the of area of the blade section. shear modulus; It is the torsional stiffness or torsional constant; and Ip is the polar moment It is important to note that in practice, mixing of modes can occur, and accordingly, of area of the blade section. various researchers have executed studies on experimental and theoretical evaluations It is important to note that in practice, mixing of modes can occur, and accordingly, of flexural-torsional vibration analysis, taking into account the coupling of flexural and various researchers have executed studies on experimental and theoretical evaluations of torsional modes, e.g., [24,27,28,31,33,34]. The present authors have only dealt with pure flexural-torsional vibration analysis, taking into account the coupling of flexural and tor- bending and torsional vibration modes, keeping in mind the simplest possible analytical sional modes, e.g., [24,27,28,31,33,34]. The present authors have only dealt with pure description mode. bending and torsional vibration modes, keeping in mind the simplest possible analytical In the literature, basic concepts are available for the analytical treatment of the vibra- description mode. tion of beams/blades affected by centrifugal force field due to their rotation. Such concepts In the literature, basic concepts are available for the analytical treatment of the vibra- regard both untwisted [31] and twisted [24,28,29] geometries. As discussed in [25], the tion of beams/blades affected by centrifugal force field due to their rotation. Such concepts centrifugal force originating from the rotation of the blades has a stiffness-increasing effect, regard both untwisted [31] and twisted [24,28,29] geometries. As discussed in [25], the i.e., it tends to moderate the inclination of the blade to vibrate. At the present state of centrifugal force originating from the rotation of the blades has a stiffness-increasing ef- research, the authors neglect the mechanical effect of the centrifugal field, for the following fect, i.e., it ten reasons: (a)dThe s to moder centrifugal ate the inc field lin tends ation of to be the blad of moderate e to vibrate significance . At the presen in the low-speed t state of rese fan arch, t blades he discussed authors neg her lec ein; t the mecha and (b) n the icaintention l effect of t is he cent to make rifua ga pessimistic—i.e., l field, for the follsafety- ow- increasing—preliminary design approach via neglecting the stiffness-increasing trend due ing reasons: (a) The centrifugal field tends to be of moderate significance in the low-speed fan to bl the ades centrifugal discussed h field. erei Blade n; antwist d (b) t tends he int to ent reduce ion is t the o mak bending e a peigenfr essimist equencies, ic—i.e., sa as fet FEM y- computations (not presented herein) demonstrate. Therefore, for twisted blades, the critical increasing—preliminary design approach via neglecting the stiffness-increasing trend frequencies of excitation tend to be shifted toward lower values. At the present state of due to the centrifugal field. Blade twist tends to reduce the bending eigenfrequencies, as research, the authors neglect the mechanical effect of blade twist. The reasonability of such FEM computations (not presented herein) demonstrate. Therefore, for twisted blades, the neglect is commented on later on. The concerted review of the effects of centrifugal force critical frequencies of excitation tend to be shifted toward lower values. At the present field and blade twisting, or taking into account the mixing of the pure vibration modes, is state of research, the authors neglect the mechanical effect of blade twist. The reasonability planned to be the subject of future research. of such neglect is commented on later on. The concerted review of the effects of centrifugal Even mechanical or fluid mechanical excitations can unavoidably induce the vibration force field and blade twisting, or taking into account the mixing of the pure vibration of the axial flow fan blade to a certain extent, as well as being a source of vibration in the modes, is planned to be the subject of future research. structure on which it is installed. Such excitation effect may derive from the interaction of Even mechanical or fluid mechanical excitations can unavoidably induce the vibra- the fan blades with the wake developing behind the elements placed upstream of the rotor tion of the axial flow fan blade to a certain extent, as well as being a source of vibration in e.g., supporting struts, inlet guide vanes, or even from the discussed PVS phenomenon. the structure on which it is installed. Such excitation effect may derive from the interaction For instance, corresponding to the pressure rise, a steady mean lift force acts on the blade. of the fan blades with the wake developing behind the elements placed upstream of the However, the lift force also has a varying component e.g., the fluctuating force due to PVS. rotor e.g., supporting struts, inlet guide vanes, or even from the discussed PVS phenom- As detailed by [25,35], both forces produce a bending moment, resulting in vibration. If the enon. For instance, corresponding to the pressure rise, a steady mean lift force acts on the dominant frequency of PVS coincides with an eigenfrequency of the blade, resonance may blade. However, the lift force also has a varying component e.g., the fluctuating force due occur, and the intensity of the vibration can only be limited if the mechanical structure is stiff enough or damped sufficiently. Therefore, it is to be treated with special care in the case of fan blades made of cambered sheet metal plates because their eigenfrequency—due to the moderate inertia of the cross-section and the resulting lower stiffness—is lower compared to that of the profiled blades [25]. Int. J. Turbomach. Propuls. Power 2022, 7, 2 5 of 23 Even though fans commonly operate with mechanical stresses far below the capacity of their material—and thus the vibration of the fan blade due to resonance does not lead to the fracture of the mechanical structure—, in the presence of the corresponding stresses, which reach a maximum near the blade root, there can be a risk of fatigue fracture. These stresses combine with the centrifugal ones; therefore, a mixed type, alternating stress condition appears, which must be considered in the design of the fan. Furthermore, the rotor may become imbalanced due to the deformation patterns associated with each resonance frequency of the blade, forcing the shaft to bend, escalating the initial imbalance and bending. If the excitation is sufficiently intense, resulting in a vibration of large amplitude, one of the fan blades can rub into the duct walls leading to rotor imbalance or the breaking down of the blades. 2.1. Analytical Treatment 2.1.1. Bending Modes In order to provide a straightforward and comprehensive approach in preliminary design, avoiding any need for numerical computation at the present phase of research and as a solutions of Equation (3), the following analytically expressed eigenfrequencies are considered for the bending modes of an arbitrary prismatic rod, as specified in the literature [26]: K E I B,i f = (6) B,i l A r Namely, the simple analytical formula above can easily be extended to higher-order bending modes by substituting the appropriate K constant into Equation (6). The values B,i of K for the first three bending modes are K = 0.560, K = 3.506 and K = 9.819, B,i B,1 B,2 B,3 respectively. 2.1.2. Torsional Modes By solving the expression in Equation (5), the eigenfrequency for torsional modes can be obtained using the analytical formula e.g., in [26], as follows: i 0.5 G I f = (7) T,i 2l r I b p In Equation (7) every variable can be determined—except I —by utilizing the material and geometrical parameters of the blade. To calculate the I torsional constant for a flat plate, the following relation is given in [26]: I = K ct (8) t t where c is the chord length (width of the plate); t is the thickness (height of the plate); and K 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 [36–38], the torsional constant of a thin-walled open tube cross-section of uniform thickness can be expressed as: I = Ut (9) where U is the length of the midwall perimeter, shown dashed later in Figure 5. Int. J. Turbomach. Propuls. Power 2022, 7, 2 6 of 23 2.2. Finite Element Method (FEM) The FEM is a useful and generally accepted tool for solving engineering problems numerically, such as the modal analysis of rotor blades [39–41]. The desired mechanical characteristics can be computed by dividing an arbitrary mechanical structure into simple geometric shapes and defining the material properties and governing connections among these elements. In this study, the frequency analysis of a low speed axial fan blade is carried out using a commercial FEM software program ANSYS Mechanical APDL 2019 R3. 2.2.1. Geometry, Materials, and Elements In the present paper, the hub is assumed to be rigid in comparison to the fan blade, thus all degrees of freedom of the blades at the hub, i.e., at the blade root, are constrained. Therefore, only one segment of the axial flow fan composed of a single blade without a hub part was examined. In accordance with the later investigated PVS-affected rotor blade profile of the circular-arc-cambered plate of 8% relative curvature and the dimensional and dimensionless values of Tables 3 and 5, the necessary geometrical characteristics for 3D modeling are summarized in Table 1. The reason for choosing this blade profile geometry will be explained in detail later in Section 3.2. For the FEM case study, the fan blade is made of structural steel with a density of 7850 kg/m , a Young’s modulus of 200 GPa, and a Poisson’s ratio (n ) of 0.3. Table 1. Blade geometrical and material parameters for the FEM case study. l c t h r E n 264 mm 120 mm 2.40 mm 9.60 mm 7850 kg/m 200 GPa 0.30 According to the literature [30,39], SHELL 181 element is suitable for analyzing thin to moderately-thick shell structures related to turbomachinery blades, such as wind turbine and axial flow fan blades. Hence, this type of element is applied to model the fan blade with 124 nodes in the axial direction and 256 nodes in the radial direction. Basically, SHELL 181 is a four-node element with six degrees of freedom at each node: translations in the x, y, and z directions and rotations about the x, y, and z-axes. However, it should be mentioned that the same solution could be obtained by using the 20-node Hex20 (SOLID 186) element of Ansys Workbench 2019 R3. 2.2.2. Mesh Convergence A mesh convergence study was performed to assure the optimum mesh number in terms of computational accuracy. The first three bending (B) and torsional (T) eigenfrequen- cies of the fan blade are computed in several mesh sizes (the element number varies from 258 to 507,904). By increasing the number of elements, the first bending eigenfrequency increases slightly and then becomes nearly constant. For higher-order bending and the torsional mode, similar effects can be observed. As shown in Figure 3, 31,744 elements are appropriate to get a sufficiently accurate solution. Nevertheless, it should be mentioned that even a model with a lower number of elements is sufficient for the third-octave band prediction of the first bending eigenfrequency (see Appendix A). Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 7 of 23 Int. J. Turbomach. Propuls. Power 2022, 7, 2 7 of 23 Figure 3. Mesh convergence study. Figure 3. Mesh convergence study. 2.3. Comparison of the Results 2.3. Comparison of the Results In Table 2, the first three analytical bending (B) and torsion (T) eigenfrequencies In Table 2, the first three analytical bending (B) and torsion (T) eigenfrequencies were were calculated based on Equations (5), (7) and (8), and compared to the eigenfrequency calculated based on Equations (5), (7) and (8), and compared to the eigenfrequency ob- obtained by means of FEM. The first column indicates the vibration mode case under tained by means of FEM. The first column indicates the vibration mode case under dis- discussion. In the second and the fourth columns, FEM and the analytical eigenfrequencies cussion. In the second and the fourth columns, FEM and the analytical eigenfrequencies are summarized, respectively. In the third and the fifth columns, the appropriate band are summarized, respectively. In the third and the fifth columns, the appropriate band number of the third-octave band resolution is listed in accordance with Appendix A. The number of the third-octave band resolution is listed in accordance with Appendix A. The “Discrepancy” column contains the relative discrepancy of f in comparison to f . analytical FEM “Discrepancy” column contains the relative discrepancy of fanalytical in comparison to fFEM. Table 2. Geometrical and material parameters of the blade. Table 2. Geometrical and material parameters of the blade. f 1/3 Octave f 1/3 Octave Discrepancy FEM analytical fFEM 1/3 Octave Band fanalytical 1/3 Octave Band Discrepancy Mode Mode [Hz] Band No. [Hz] Band No. [%] [Hz] No. [Hz] No. [%] B,1 116.5 11 116.1 11 0.3 B,1 116.5 11 116.1 11 0.3 B,2 595.2 18 727.6 19 18.2 B,2 595.2 18 727.6 19 18.2 B,3 1146.7 21 2037.8 23 43.7 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,1 140.1 11 117.4 11 19.4 T,3 949.6 20 586.9 18 61.8 T,2 472.8 17 352.1 15 34.3 T,3 949.6 20 586.9 18 61.8 From Table 2, it can be observed that the analytical treatment for the first bending (B,1) eigenfrequency is in good agreement with the FEM result. Nevertheless, for the From Table 2, it can be observed that the analytical treatment for the first bending higher-order bending and the torsional modes, the relative discrepancy between analytical (B,1) eigenfrequency is in good agreement with the FEM result. Nevertheless, for the and FEM results becomes greater and tends to increase with the increasing of the order of higher-order bending and the torsional modes, the relative discrepancy between analyti- the modes. A possible explanation for this is the partial violation of the briefly presented cal and FEM results becomes greater and tends to increase with the increasing of the order Euler–Bernoulli thin beam theory. Hence, the AR of the beam is equal to 2.2; namely, the of the modes. A possible explanation for this is the partial violation of the briefly presented beam is not considered to be slender (AR > 10). Euler–Bernoulli thin beam theory. Hence, the AR of the beam is equal to 2.2; namely, the Although there have been alternative methodologies to obtain a more accurate an- beam is not considered to be slender (AR > 10). alytical prediction for short beams (AR < 10), e.g., Timoshenko’s beam theory [31], the Although there have been alternative methodologies to obtain a more accurate ana- present paper focuses on Euler–Bernoulli’s beam theory, due to the following reasons. lytical prediction for short beams (AR < 10), e.g., Timoshenko’s beam theory [31], the pre- (a) On the one hand, the authors aim to create a closed analytical formula with the most sent paper focuses on Euler–Bernoulli’s beam theory, due to the following reasons. (a) On straightforward possible analytical description in mind. (b) On the other hand, as shown the one hand, the authors aim to create a closed analytical formula with the most straight- in [31,42], the relative discrepancy between the analytical first bending eigenfrequencies, forward possible analytical description in mind. (b) On the other hand, as shown in calculated based on the two different theories, is less than 5–10%, which, fitting for point [31,42], the relative discrepancy between the analytical first bending eigenfrequencies, cal- (a), is considered to be an acceptable approximation. Based on this and the highlighted role culated based on the two different theories, is less than 5–10%, which, fitting for point (a), of the first bending mode detailed in Section 3.1, the confirmation of the analytical model is considered to be an acceptable approximation. Based on this and the highlighted role by FEM is of primary significance for the first bending mode only and is of secondary of the first bending mode detailed in Section 3.1, the confirmation of the analytical model importance for the other modes. by FEM is of primary significance for the first bending mode only and is of secondary Despite the simplification assumption discussed at the beginning of the chapter (un- importance for the other modes. twisted and non-rotating fan blade), supplementary FEM case studies were carried out Despite the simplification assumption discussed at the beginning of the chapter (un- for the twisted and rotating blade as well. The FEM results demonstrated that both the twisted and non-rotating fan blade), supplementary FEM case studies were carried out blade twisting and the presence of a centrifugal force field have a bending eigenfrequency- for the twisted and rotating blade as well. The FEM results demonstrated that both the reducing effect, providing an upper estimation for the first bending eigenfrequency of the blade twisting and the presence of a centrifugal force field have a bending eigenfrequency- blade. As the later calculation example illustrates, even in the case of the untwisted and Int. J. Turbomach. Propuls. Power 2022, 7, 2 8 of 23 stationary fan blade, only impractically low tip speeds would result in the coincidence of the first bending eigenfrequency and PVS frequency. Therefore, the simplified fan blade model applied by the present authors can be considered as a pessimistic design scenario. 3. Blade Mechanics: An Exemplary Case Study 3.1. The Importance of the First-Order Bending Mode The risk in fan operation due to blade vibration is simultaneously viewed in the present paper from the following two perspectives, which are related to each other. (a) The risk of instantaneous reduction of the gap between the blade tip and the casing. From the perspective of the tip gap reduction, the vibration modes exhibiting monotonously increasing deformation amplitude along the blade height are considered the riskiest. These are the first bending and first torsional modes. (b) The risk of instantaneous rotor imbalance. This occurs in cases when the weight point of the entire blade is displaced in the transversal direction from its original position, fitting on the (approximately) radial blade stacking line, for which the rotor has originally been balanced. From the perspective of rotor imbalance, none of the torsional modes are considered to be risky, as they tend to leave the weight point of the entire blade (approximately) in its original position. In addition, according to their wavy vibration pattern, higher-order bending modes are considered to exhibit only a moderate transversal displacement of the blade weight point. Therefore, among the various vibrational modes caused by PVS excitation, the first bending mode is judged to be the most critical, and the higher-order bending modes and torsional modes are considered of secondary significance. Consequently, out of the an- alytical solutions in Equation (5), the “first-order bending mode” is taken herein as an illustrative example of the analytical treatment elaborated upon by the authors. Neverthe- less, the reasonability of this choice is justified and supported by the subsequent examples, as follows. The impact of PVS on both noise and vibration is presumed by the present authors to be pronounced when extensive and coherent vortices are shed with uniform frequency along a dominant portion of the blade span. In this case, PVS is assumed to exhibit pressure fluctuations over the blade suction and pressure surfaces, causing chord-normal forces. Assuming spanwise and spatially coherent shed vortices, the resultant fluctuating forces are in phase over the entire span. Furthermore, PVS and the associated forces may occur farther upstream of the TE (cf. [12]). Such a PVS-induced excitation likely triggers the first-order bending mode. An upstream stator, e.g., the nearly radially aligned supporting struts located upstream of the rotor, is able to cause wake-blade—also known as “rotor-stator”—interaction, in which the upstream wakes of the stator are swept downstream into the axial flow fan blade- row. The wakes are parallel with the relative velocity (w), thus the interaction manifests itself as a nearly simultaneous, spatially coherent aerodynamic excitation along the entire blade span. Hence, the fan blades are acted by chord-normal fluctuating force, resulting in a spatially coherent bending moment rather than torsional. In addition, in [25] as a general approach, the frequency of the first bending mode of the blades is to be kept far away from the frequencies of excitation. As it was just mentioned, elements located upstream of the rotor—supporting struts and inlet guide vanes—cause rotor–stator interaction as aerodynamic excitation to the rotor blades, occur- ring at a frequency of rotational frequency multiplied by the number of upstream elements. On the other hand, such excitation effects correspond to rotor imbalance (“shaker effect”) appearing as mechanical excitation at the rotational frequency of the fan. Moreover, as illustrated in [5], the fluctuation of chord-normal force due to PVS may lead to a variance of the lift coefficient in the order of a magnitude of 10 percent of the temporal mean value. As demonstrated in [35], the varying component of lift force causes a bending moment on the blade. When extensive and coherent vortices are shed with uniform frequency along a dominant portion of the blade span, they represent spatially coherent elemental aerodynamic excitation forces, being in phase over the elemental blade Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 9 of 23 a bending moment on the blade. When extensive and coherent vortices are shed with uni- Int. J. Turbomach. Propuls. Power 2022, 7, 2 9 of 23 form frequency along a dominant portion of the blade span, they represent spatially co- herent elemental aerodynamic excitation forces, being in phase over the elemental blade sections, and thus, integrated into a pronounced overall bending moment. The phase sections, and thus, integrated into a pronounced overall bending moment. The phase identity of the elemental excitation forces along the blade span matches with the phase identity of the elemental excitation forces along the blade span matches with the phase identity of blade deformation in the first bending mode of the blade, if the frequency of identity of blade deformation in the first bending mode of the blade, if the frequency of PVS matches with the eigenfrequency related to the first bending mode. PVS matches with the eigenfrequency related to the first bending mode. 3.2. Eigenfrequency 3.2. Eigenfrequency In order to build up a straightforward model for the interaction of the blade and the In order to build up a straightforward model for the interaction of the blade and the fluctuating aerodynamic force due to PVS, some simplifications must be introduced. First fluctuating aerodynamic force due to PVS, some simplifications must be introduced. First of all, a circular-arc-cambered plate of 8% relative curvature is chosen as a blade profile of all, a circular-arc-cambered plate of 8% relative curvature is chosen as a blade profile for for several reasons: several reasons: (a) At moderate Reynolds numbers and angles of attack (α), the cambered plate pro- (a) At moderate Reynolds numbers and angles of attack (a), the cambered plate produces duces reasonably high CL, that is comparable with an airfoil profile, i.e., RAF-6E [43], reasonably high C , that is comparable with an airfoil profile, i.e., RAF-6E [43], thus thus enabling the design of blades of relatively high specific performance, i.e., utiliz- enabling the design of blades of relatively high specific performance, i.e., utilizing the ing the loading capability of the blade sections. loading capability of the blade sections. (b) At 8% relative curvature, the lift-to-drag (LDR) is near the maximum among the cam- (b) At 8% relative curvature, the lift-to-drag (LDR) is near the maximum among the bered plates of various relative camber, thus enabling the design for reasonably high cambered plates of various relative camber, thus enabling the design for reasonably efficiency [44]. high efficiency [44]. (c) (c) In accord In accordance ance with the with the afor afor ementioned p ementioned rpractical actical aspec aspects, ts, it is it a c is a amb camber ered plate o ed plate f 8% of relative camber for which hot-wire measurement data are made available by the pre- 8% relative camber for which hot-wire measurement data are made available by the sent authors present authors on P on VPVS S at different at different free free-str -stream eam ve velocities locities and ang and angles les of at of attack tack [ [1 13 3]. ]. As As per per the illustration, the illustration, F Figur igure 4 shows e 4 shows C CL, , C CD, and , and LD LDR R va values lues a as s a fu a function nction of the of the angl angle e L D of attack for the 8% cambered plate. of attack for the 8% cambered plate. Figure 4. Lift and drag coefficients and lift-to-drag ratios as a function of angle of attack for 8% Figure 4. Lift and drag coefficients and lift- 5 to-drag ratios as a function of angle of attack for 8% cambered plate at Re = 3  10 measured by Wallis [44]. cambered plate at Rec = 3 × 10 measured by Wallis [44]. Secondly, the fan blade is presumed to consist of geometrically identical blade sections Secondly, the fan blade is presumed to consist of geometrically identical blade sec- along the full span. This means that the blade chord c, plate thickness t, and height of the tions along the full span. This means that the blade chord c, plate thickness t, and height camber line h are constants (Figure 5). The measurement data presented by [44] and shown of the camber line h are constants (Figure 5). The measurement data presented by [44] and in Figure 4, are related to a relative thickness (t/c) of 2%. This value is representative in fan shown in Figure 4, are related to a relative thickness (t/c) of 2%. This value is representa- manufacturing; therefore, t/c is fixed at 2% for the present investigations. tive in fan manufacturing; therefore, t/c is fixed at 2% for the present investigations. Int. J. Turbomach. Propuls. Power 2022, 7, 2 10 of 23 Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 10 of 23 Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 10 of 23 Figure 5. Circular-arc-cambered plate (8% relative camber). Figure 5. Circular-arc-cambered plate (8% relative camber). Figure 5. Circular-arc-cambered plate (8% relative camber). Finally, as discussed earlier, it is assumed that all degrees of freedom of the blades at Finally, as discussed earlier, it is assumed that all degrees of freedom of the blades at Finally, as discussed earlier, it is assumed that all degrees of freedom of the blades at the hub, i.e., at the blade root, are constrained. This is a reasonable approximation; e.g., the hub, i.e., at the blade root, are constrained. This is a reasonable approximation; e.g., if the hub, i.e., at the blade root, are constrained. This is a reasonable approximation; e.g., if if sheet metal blades connect to the hub with a welded joint, or, in the case of polymer sheet metal blades connect to the hub with a welded joint, or, in the case of polymer ma- sheet metal blades connect to the hub with a welded joint, or, in the case of polymer ma- material, if the entirety of the hub and blading assembly is injection-molded as a single terial, if the entirety of the hub and blading assembly is injection-molded as a single prod- terial, if the entirety of the hub and blading assembly is injection-molded as a single prod- product. The simplifications above enable us to model the blade as a cantilever beam uct. The simplifications above enable us to model the blade as a cantilever beam (or uct. The simplifications above enable us to model the blade as a cantilever beam (or (or clamped beam) subjected to free vibration. In accordance with Section 3.1, the lowest clamped beam) subjected to free vibration. In accordance with Section 3.1, the lowest ei- clamped beam) subjected to free vibration. In accordance with Section 3.1, the lowest ei- eigenfrequency, related to the first-order bending mode, can be calculated as follows: genfrequency, related to the first-order bending mode, can be calculated as follows: genfrequency, related to the first-order bending mode, can be calculated as follows: 0.56 I E 0.56 I E 0.56 I E f = (10) B,1 f = f = B,1 (10) 2 2 (10) B,1 2 l A r l A ρ b l A ρ The The la last st term term on on the right- the right-hand hand side side of of Eq Equation uation (10), as (10), as the the squar square e r root of the sp oot of the specific ecific The last term on the right-hand side of Equation (10), as the square root of the specific modulus modulus E E/ /ρrb, is termed h , is termed her erein the w ein the wave ave pr propagation opagation speed, an speed, and d denoted denoted as as aab.. This This is is modulus E/ρb b, is termed herein the wave propagation speed, and denoted as a bb. This is actually the acoustic wave propagation speed in a long one-dimensional fictitious beam actually the acoustic wave propagation speed in a long one-dimensional fictitious beam 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. made of the blade material. The resultant swinging pattern is shown in Figure 6. made of the blade material. The resultant swinging pattern is shown in Figure 6. Figure 6. Cantilever beam subjected to free vibration (perpendicular to the blade chord). Figure 6. Cantilever beam subjected to free vibration (perpendicular to the blade chord). Figure 6. Cantilever beam subjected to free vibration (perpendicular to the blade chord). 3.3. Second Moment of Area 3.3. Second Moment of Area 3.3. Second Moment of Area The purpose of the present section is to demonstrate how the I/A term can be related The purpose of the present section is to demonstrate how the I/A term can be related The purpose of the present section is to demonstrate how the I/A term can be related to the basic geometrical characteristics of the circular-arc-cambered plate blade (c, h/c, to the basic geometrical characteristics of the circular-arc-cambered plate blade (c, h/c, t/c), to the basic geometrical characteristics of the circular-arc-cambered plate blade (c, h/c, t/c), t/c), creating a direct relationship between the first eigenfrequency and the geometrical creating a direct relationship between the first eigenfrequency and the geometrical param- creating a direct relationship between the first eigenfrequency and the geometrical param- parameters of the blade. However, no closed analytical relationship exists in the literature eters of the blade. However, no closed analytical relationship exists in the literature for eters of the blade. However, no closed analytical relationship exists in the literature for for such purpose. Therefore, an alternative method must be found to express the second such purpose. Therefore, an alternative method must be found to express the second term such purpose. Therefore, an alternative method must be found to express the second term term of the right-hand side of Equation (10) with the use of blade geometrical parameters, of the right-hand side of Equation (10) with the use of blade geometrical parameters, such of the right-hand side of Equation (10) with the use of blade geometrical parameters, such such as c, h, and t. As a first step, the second moment of area of a cambered plate and a flat as c, h, and t. As a first step, the second moment of area of a cambered plate and a flat plate as c, h, and t. As a first step, the second moment of area of a cambered plate and a flat plate plate with same geometrical parameter (t, c) were compared to each other. with same geometrical parameter (t, c) were compared to each other. with same geometrical parameter (t, c) were compared to each other. The quotient of the second moment of area and the cross-section, in case of a flat The quotient of the second moment of area and the cross-section, in case of a flat plate The quotient of the second moment of area and the cross-section, in case of a flat plate plate is: is: is: 3 2 (I /A) = (ct /12)/(ct) = t /12 (11) flat 3 2 3 2 () I A = (ct /12) (ct) = t 12 (11) () I A = (ct /12) (ct) = t 12 (11) flat For a cambered plate it is: flat For a cambered plate it is: For a cambered plate it is: (I /A) = K (I /A) , where K = K (t/c, h/c) (12) 1 1 1 cambered flat () I A = K() I A , where K = K (t/c, h/c) (12) () I A = K() I A , where K = K (t/c, h/c) 1 1 1 (12) cambered flat cambered 1 flat 1 1 I was obtained for the cambered plate-section using an analytical integration cambered process known from basic solid-state mechanics [45]. As background information for the Icambered was obtained for the cambered plate-section using an analytical integration Icambered was obtained for the cambered plate-section using an analytical integration reader, the values of K are presented in Figure 7 for representative relative thickness and process known from basic solid-state mechanics [45]. As background information for the process known from basic solid-state mechanics [45]. As background information for the relative camber (h/c) values. For the fitted curves in Figure 7, K was calculated for fixed t/c reader, the values of K1 are presented in Figure 7 for representative relative thickness and reader, the values of K1 are presented in Figure 7 for representative relative thickness and values for uniform steps of 0.01 h/c over the entire h/c range. In the blade design presented relative camber (h/c) values. For the fitted curves in Figure 7, K1 was calculated for fixed relative camber (h/c) values. For the fitted curves in Figure 7, K1 was calculated for fixed later, K (t/c = 0.02; h/c = 0.08) = 18.07 was used, in accordance with the previously selected t/c values for uniform steps of 0.01 h/c over the entire h/c range. In the blade design pre- t/c values for uniform steps of 0.01 h/c over the entire h/c range. In the blade design pre- blade geometrical parameters. sented later, K1 (t/c = 0.02; h/c = 0.08) = 18.07 was used, in accordance with the previously sented later, K1 (t/c = 0.02; h/c = 0.08) = 18.07 was used, in accordance with the previously selected blade geometrical parameters. selected blade geometrical parameters. Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 11 of 23 Substituting Equations (11) and (12) into Equation (10), the first bending eigenfre- quency can be expressed as follows: 0.56 t 0.56 (13) f = K a = K t a B,1 1 b b b 2 2 l 12 l Int. J. Turbomach. Propuls. Power 2022, 7, 2 11 of 23 where Kb = Kb (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. Figure Figure 7. 7. The The v value alue of of K K1 as a f as a function unction of relat of relative ive camber camber for for various various relative thicknesses. relative thicknesses. Substituting Equations (11) and (12) into Equation (10), the first bending eigenfre- 4. PVS-Affected Rotor: An Exemplary Case Study quency can be expressed as follows: The aim of this section is to outline a design case study, resulting in a rotor suspected to be unfavorable in terms of noise and vibration due to PVS. Although several unfavor- 0.56 t 0.56 able design cases of such kind can be inte f = K ntionaally genera = K ted, thi t a s paper intends to present (13) B,1 1 b b b 2 2 l l a single, representative design example. The resultant rotor will be termed herein the PVS- where K = K (t/c = 0.02; h/c = 0.08) = 1.23 is the blade mechanics coefficient. Case studies affected rotor. Such a design aims to serve as a basis for the realization of a case study b b considering other t/c and h/c values can be carried out using Figure 7. rotor that is expected to exhibit remarkable and experimentally well-detectable narrow- band signatures of PVS. By such means, the orders of the magnitude of harmfulness of 4. PVS-Affected Rotor: An Exemplary Case Study PVS-related effects are to be quantified in future experiments on the PVS-affected rotor, The aim of this section is to outline a design case study, resulting in a rotor suspected in comparison with other rotors of comparative design. The features of the PVS-affected to be unfavorable in terms of noise and vibration due to PVS. Although several unfavorable rotor are as follows: design cases of such kind can be intentionally generated, this paper intends to present (a) The rotor blading tends to exhibit a PVS of spanwise constant frequency, and thus, is a single, representative design example. The resultant rotor will be termed herein the theoretically presumed to realize large-scale, spatially coherent vortices over the PVS-affected rotor. Such a design aims to serve as a basis for the realization of a case dominant portion of the blade span. Such coherent vortices are assumed to cause study rotor that is expected to exhibit remarkable and experimentally well-detectable spatially correlated, narrowband noise, as well as mechanical excitation over the narrowband signatures of PVS. By such means, the orders of the magnitude of harmfulness dominant part of span at a given frequency. The condition of spanwise constant PVS of PVS-related effects are to be quantified in future experiments on the PVS-affected rotor, frequency is therefore applied herein in a pessimistic aspect, although it is noted that in comparison with other rotors of comparative design. The features of the PVS-affected the signature of PVS noise was observed by [19] even when PVS was confined to rotor are as follows: ~10% span near the tip of a single blade. (a) The rotor blading tends to exhibit a PVS of spanwise constant frequency, and thus, (b) The latter is presumed to provoke blade vibration, if the frequency of PVS coincides is theoretically presumed to realize large-scale, spatially coherent vortices over the with the frequency of the first bending mode of blade vibration. dominant portion of the blade span. Such coherent vortices are assumed to cause spatially correlated, narrowband noise, as well as mechanical excitation over the 4.1. Aerodynamics: Blade Design dominant part of span at a given frequency. The condition of spanwise constant PVS To be able to design a fan for which the spanwise constancy of PVS frequency is ful- frequency is therefore applied herein in a pessimistic aspect, although it is noted that filled—thus satisfying Equation (14), presented later—first, typical dimensionless design the signature of PVS noise was observed by [19] even when PVS was confined to ~10% and geometrical characteristics, representative of propeller-type fans, were systematically span near the tip of a single blade. gathered and summarized in Table 3. Here, the authors stress that, in order to guarantee (b) The latter is presumed to provoke blade vibration, if the frequency of PVS coincides the validity of Equation (9), the blade chord along the blade span was fixed, so c(rb) = with the frequency of the first bending mode of blade vibration. 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 Cor- 4.1. Aerodynamics: Blade Design dier diagram [46] for turbomachines of favorable efficiency, axial flow fans have the spe- To be able to design a fan for which the spanwise constancy of PVS frequency is cific diameter and the specific speed within the approximate ranges of 1 ≤ δ ≤ 1.5 and 2 ≤ fulfilled—thus satisfying Equation (14), presented later—first, typical dimensionless design σ ≤ 3, respectively. These ranges correspond to the global total pressure rise coefficient and geometrical characteristics, representative of propeller-type fans, were systematically and flow coefficient within the ranges of 0.05 ≤ Ψt ≤ 0.25 and 0.1 ≤ Φ ≤ 0.5, respectively. gathered and summarized in Table 3. Here, the authors stress that, in order to guarantee the validity of Equation (9), the blade chord along the blade span was fixed, so c(r ) = 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  d  1.5 and 2  s  3, respectively. These ranges correspond to the global total pressure rise coefficient and flow coefficient within the ranges of 0.05  Y  0.25 and 0.1  F  0.5, respectively. Thus, values in Table 3 fit to the Cordier diagram well. Regulation 327/2011/EU5 issued energy Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 12 of 23 Int. J. Turbomach. Propuls. Power 2022, 7, 2 12 of 23 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 effi- efficiency requirements regarding fans in the EU [47], driven by motors with an electric ciency of an axial fan is 0.5 ≤ ηt ≤ 0.6 depending on the arrangement and input power. input power between 125 W and 500 kW. According to this, the target total efficiency of an Considering a more economical operation, we are moving toward higher efficiency levels; axial fan is 0.5  h  0.6 depending on the arrangement and input power. Considering therefore, ηt ≈ 0.7 is chosen. a more economical operation, we are moving toward higher efficiency levels; therefore, h  0.7 is chosen. Table 3. Geometrical and material parameters of the blade for the PVS-affected rotor. Table 3. Geometrical and material parameters of the blade for the PVS-affected rotor. c/Dtip ν AR Ψt Φ N ηt δ σ 0.133 0.415 2.22 0.143 0.309 5 0.700 1.11 2.39 c/D n AR Y F N h d s tip t t 0.133 0.415 2.22 0.143 0.309 5 0.700 1.11 2.39 The first two parameters as well as N in Table 3 define the blade solidity from hub to tip. When calculating the solidity (c/s), it is found that it is below 0.7 along the entire radius The first two parameters as well as N in Table 3 define the blade solidity from hub of the blade even at the hub, as can be observed in Figure 8. Therefore, according to [46] to tip. When calculating the solidity (c/s), it is found that it is below 0.7 along the entire measurement data on isolated blade profiles—such as data included in Figure 4 in the radius of the blade even at the hub, as can be observed in Figure 8. Therefore, according case of a 8% cambered plate—can be applied in the present design of the PVS-affected to [46] measurement data on isolated blade profiles—such as data included in Figure 4 in rotor. In addition to the aerodynamic data in Figure 4, the empirical data on PVS in [5] the case of a 8% cambered plate—can be applied in the present design of the PVS-affected and [13] are also related to isolated profiles, thus fitting to the low-solidity approach uti- rotor. In addition to the aerodynamic data in Figure 4, the empirical data on PVS in [5,13] lized herein. are also related to isolated profiles, thus fitting to the low-solidity approach utilized herein. 0.75 0.60 0.45 0.30 0.15 0.00 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Figure 8. Blade solidity as a function of dimensionless radius (R = r/r ). tip Figure 8. Blade solidity as a function of dimensionless radius (R = r/rtip). In what follows, preliminary design efforts are made for obtaining a rotor blading exhibiting PVS of spanwise constant frequency in order to fulfill the pessimistic condi- In what follows, preliminary design efforts are made for obtaining a rotor blading tion outlined at the end of the Introduction. The PVS frequency can be expressed from exhibiting PVS of spanwise constant frequency in order to fulfill the pessimistic condition Equation (2) as follows: outlined at the end of the Introduction. The PVS frequency can be expressed from Equa- U (r ) tion (2) as follows: f (r ) = St = const (14) PVS b b(r ) U (r ) 0 b A nearly constant value off St* (r ) = 0.19 St was found = const for a 8% cambered plate for various (14) PVS b b(r ) Reynolds numbers and angles of attack in the present authors’ previous measurement campaign [13]. Keeping the uncertainty of the measurement-based St* data in mind, this A nearly constant value of St* ≈ 0.19 was found for a 8% cambered plate for various value is in fair agreement with the literature-based data in Equation (2). St* = 0.19 was used Reynolds numbers and angles of attack in the present authors’ previous measurement for the calculation presented in the paper. In order to clarify the trend of the free-stream campaign [13]. Keeping the uncertainty of the measurement-based St* data in mind, this velocity, the velocity vectors are shown in Figure 9. Based on Figure 9, the square of the value is in fair agreement with the literature-based data in Equation (2). St* = 0.19 was free-stream velocity can be written as follows [46]: used for the calculation presented in the paper. In order to clarify the trend of the free- stream velocity, the velocity vectors are shown  in Figure 9. Based on Figure 9, the square Dc (r ) 2 2 b of the free-stream velocity can be written as follows [46]: U (r ) = c (r ) + u(r ) (15) 0 x b b b Δc (r ) 2 2   u b U (r ) = c (r ) + u (r ) −   (15) where c is the axial velocity component; u is the circumferential velocity; and Dc is the x 0 b x b b u   increase of tangential velocity due to the rotor. where cx is the axial velocity component; u is the circumferential velocity; and Δcu is the increase of tangential velocity due to the rotor. c/s Int. J. Turbomach. Propuls. Power 2022, 7, 2 13 of 23 Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 13 of 23 Figure 9. Velocity triangles. Figure 9. Velocity triangles. From Equation (15) the square of the dimensionless free-stream velocity can be ex- From Equation (15) the square of the dimensionless free-stream velocity can be ex- pr pressed essed as as fo follows: llows: U (r )  ψ (R) U (r ) y (R) 2 t,is 2 0 b 0 b t,is 2 2 = ϕ (R) + R −  ≡ W (R) = j (R) + R  W (R ) (16) (16)   u 44RR tip tip   where W is the dimensionless free-stream velocity; j = c (r )/u is the local flow coefficient; where W is the dimensionless free-stream velocity; φ = cx(rb)/utip is the local flow coeffi- x b tip R is the dimensionless radius; and y is the local isentropic total pressure rise coefficient. cient; R is the dimensionless radius; and ψt,is is the local isentropic total pressure rise coef- t,is As ficient a brief . As appr a broximation, ief approxim the atcir ion, cumfer the ci ential rcumferent velocity ial v u,erlocit epresenting y u, repra esenting solid body a sor lid bod otation, y dominates in U . Therefore, U tends to approximately linearly increase with R. In a refined rotation, dominates in U0. Therefore, U0 tends to approximately linearly increase with R. 0 0 calculation, presented later, c /u and Dc /u are taken into account when obtaining U . In a refined calculation, presented x tip later, ucx/u tip tip and Δcu/utip are taken into account when 0 In order to provide a spanwise constant f , b(r ) tends to increase along the span by obtaining U0. PVS b such means that its increase matches with the spanwise increase of U . In order to provide a spanwise constant fPVS, b(rb) tends to increa 0se along the span by According to [5], the distance between vortex rows normalized by the blade chord is: such means that its increase matches with the spanwise increase of U0. According to [5], the distance between vortex rows normalized by the blade chord is: b(r ) Q(r ) b b = K (17) bc (r ) Θ c (r ) b b = K (17) c c where K*  1.2 is an empirical coefficient [5] and Q is the momentum thickness of the blade wake. According to [17,48] the drag coefficient can be written as: where K* ≈ 1.2 is an empirical coefficient [5] and Θ is the momentum thickness of the blade wake. According to [17,48] the drag coefficient can be written as: 2Q(r ) C (r ) = (18) D b 2Θ(r ) C (r ) = (18) D b Thus b can be expressed as follows: Thus b can be expressed as follows: C (r ) D b b(r ) = c K (19) C (r ) * D b b(r ) = c K (19) From Equation (19) it can be observed that a spanwise increase of b can be achieved via From Equation (19) it can be observed that a spanwise increase of b can be achieved spanwise increase of C . Based on Figure 4, C (R ) = 0.032 is chosen, which corresponds D D mid via spanwise increase of CD. Based on Figure 4, CD(Rmid) = 0.032 is chosen, which corre- to a = 6.8 . In the design of the PVS-affected rotor, the following range of C (R) was used: sponds to α = 6.8°. In the design of the PVS-affected rotor, the following range of CD(R) 0.0196  C  0.0410. Furthermore, as Figure 4 illustrates, the C data within the design D D was used: 0.0196 ≤ CD ≤ 0.0410. Furthermore, as Figure 4 illustrates, the CD data within the range are assigned to a data, and via such assignment, they also determine the design range design range are assigned to α data, and via such assignment, they also determine the for the local lift coefficient C (a). Therefore, the lift-to-drag ratio LDR(a) = C /C data are L L D design range for the local lift coefficient CL(α). Therefore, the lift-to-drag ratio LDR(α) = also obtained for the entire design range, incorporating the data at R . Thus, each of mid CL/CD data are also obtained for the entire design range, incorporating the data at Rmid. C (R ), C (R ), and LDR(R ) are available; these quantities will play an important D mid L mid mid Thus, each of CD(Rmid), CL(Rmid), and LDR(Rmid) are available; these quantities will play an role in the further investigation of the PVS-affected rotor, as presented later. important role in the further investigation of the PVS-affected rotor, as presented later. Iterative Method Iterative Method In order to provide a spanwise constant f , the nearly linear spanwise increase of PVS U is In order to matched in provide a sp blade design anwise const with the spanwise ant fPVS, the in ne crar ease ly line of C ar sp . T an o wise be able incrto ease design of U0 0 D such is ma atched fan, an in bla iterative de design wi method th the is elaborated spanwise i asnfollows, crease ofusing CD. To be the data ablein to desi Table gn su 3 as ch thea basis. fan, an Using iterative method is elabor Equation (16) as an ini ated tialas foll guess, ows, using the data the increase of tangential in Table velocity 3 as the basis due to. Using Equation (16) as an initial guess, the increase of tangential velocity due to the rotor Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 14 of 23 Int. J. Turbomach. Propuls. Power 2022, 7, 2 14 of 23 is neglected; therefore, W(R) is calculated from only the local flow coefficient φ = cx(rb)/utip and the dimensionless radius corresponding to the rigid body rotation. In the present the rotor is neglected; therefore, W(R) is calculated from only the local flow coefficient methodology, uniform axial inlet condition is assumed, cx(rb)/utip = constant. Furthermore, j = c (r )/u and the dimensionless radius corresponding to the rigid body rotation. In b tip based on [46] as an approximation, the change of meridional—i.e., axial—velocity is ne- the present methodology, uniform axial inlet condition is assumed, c (r )/u = constant. x tip glected through the rotor. The above implies that spanwise constant axial velocity is pre- Furthermore, based on [46] as an approximation, the change of meridional—i.e., axial— sumed, as a brief approximation. velocity is neglected through the rotor. The above implies that spanwise constant axial velocity As a ne is prxt s esumed, tep, ba as sea d on the val brief approximation. ue of W(Rmid)/CD(Rmid), the drag coefficients are deter- mined Asalon a next g the blade step, based span. As desc on the value ribed earlier, the of W(R )/CCD (R (R), α ),(R the ) and drag CLcoef (R) d ficients ata are as are- mid D mid determined along the blade span. As described earlier, the C (R), a(R) and C (R) data are signed to each other, via Figure 4. As a next stage of the design, Δcu/utip is expressed from D L assigned the simpli to fied work equa each other, via tion of Figur a en4 el . As ementa a next l rotor: stage of the design, Dc /u is expressed tip from the simplified work equation of an elemental rotor: c 2Δc (r ) u b C (r ) ≅ (20) L b c s(r ) 2DUc ( (rr )) b 0 bb C (r ) (20) L b s(r ) U (r ) b 0 b where s = 2rbπ/N is the blade spacing. With the knowledge of Δcu(R)/utip, the local isen- tropic total pressure rise coefficient ψt,is(R) is expressed from the Euler equation of tur- where s = 2r /N is the blade spacing. With the knowledge of Dc (R)/u , the local b u tip bomachines: isentropic total pressure rise coefficient y (R) is expressed from the Euler equation of t,is turbomachines: Δp (r ) = ρ u (r )Δc (r ) t,is b a b u b (21) Dp (r ) = r u(r )Dc (r ) (21) a u t,is b b b here Δpt,is is the isentropic total pressure rise, and ρa is the density of air. This computed here Dp is the isentropic total pressure rise, and r is the density of air. This computed t,is ψt,is(R) is then substituted into Equation (16) for calculating a new approximation of W(R) y (R) is then substituted into Equation (16) for calculating a new approximation of W(R) t,is in the next iteration loop. Fast convergence is obtained in two to three iteration loops. The 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 ψt,is(R) for the consecutive results are judged to be converged if the relative difference in y (R) for the consecutive t,is iteration steps becomes less than 2%. iteration steps becomes less than 2%. A feature of the iterative method is that ψt,is(R) and α(R) are actually the results of the A feature of the iterative method is that y (R) and a(R) are actually the results of t,is design process. The spanwise distributions of the calculated quantities are shown in Fig- the design process. The spanwise distributions of the calculated quantities are shown ure 10, where γ(R) is stagger angle measured from the circumferential direction. Annulus- in Figure 10, where g(R) is stagger angle measured from the circumferential direction. averaging of ψt,is(R) represents the global isentropic total pressure rise obtained as Ψt/ηt, Annulus-averaging of y (R) represents the global isentropic total pressure rise obtained t,is using the data in Table 3. as Y /h , using the data in Table 3. t t Figure 10. Calculated distributions y (R), a(R), and g(R) as a function of R. t,is Figure 10. Calculated distributions ψt,is(R), α(R), and γ(R) as a function of R. Figure 11 represents the proportionate three-dimensional model of the designed Figure 11 represents the proportionate three-dimensional model of the designed PVS-affected fan. The rotor has a realistic, common geometry, as it is representative of pr PVS-affected opeller-type fan axial . The flow roto industrial r has a refans. alistic, The common blade geometry, geometry isas it in accor is represen dance with tative the of propeller-type axial flow industrial fans. The blade geometry is in accordance with the fact that, as a manufacturing simplification, low-speed axial fan blades of cost-effective manufacturing fact that, as a manu can be fact made uring simpl from a plate ificati of on, low spanwise -speed ax constant ial fchor an blade d, as s they of cos aret-effe rolled ctive in such manu afact way urthat ing can be m spanwise ad constant e from a p camber late of spanw and moderate ise const twist ant chor occur d, a . s they are rolled in such a way that spanwise constant camber and moderate twist occur. Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 15 of 23 Int. J. Turbomach. Propuls. Power 2022, 7, 2 15 of 23 Figure 11. The PVS-affected fan rotor. Figure 11. The PVS-affected fan rotor. 4.2. Mechanically or Acoustically Unfavorable Design Cases 4.2. Mechanically or Acoustically Unfavorable Design Cases In order to set up mechanically unfavorable design cases, representing a coincidence In order to set up mechanically unfavorable design cases, representing a coincidence of the frequency of PVS with the frequency of the first bending mode of blade vibration, it of the frequency of PVS with the frequency of the first bending mode of blade vibration, is necessary to express U and b in Equation (14). However, b was previously expressed, it is necessary to express U0 and b in Equation (14). However, b was previously expressed, as shown in Equation (17). Therefore, it is only necessary to deal with U . Based on as shown in Equation (17). Therefore, it is only necessary to deal with U0. Based on Equa- Equations (20) and (21), the free-stream velocity is written as follows: tions (20) and (21), the free-stream velocity is written as follows: 2Dp (r ) s(r ) t,is 2Δp (b r ) s(rb) t,is b b U (r ) = (22) 0 b U (r ) = (22) 0 b r u(r ) c C (r ) a b L b ρ u(r ) c C (r ) a b L b Thus, the PVS frequency can be expressed from Equations (14), (19) and (22): Thus, the PVS frequency can be expressed from Equations (14), (19) and (22): 2D2pΔp(r(r)) s(r ) 2 t,is s(r ) 2 b b * t,is b f (r f ) (= r )S=t St (23) PVS b (23) PVS b r c C (r ) u(r ) K C (r ) c a ρ cL C (br ) u(r b ) K C D(r ) bc a L b b D b With the use of s(r )/u(r ) = (2 r /N)/(2 r n) = 1/(nN) and C = C /LDR, Equation (23) With the use of sb (rb)/u(brb) = (2 r b bπ/N)/(2 rb bπn) = 1/(nN) and CD D = CL L/LDR, Equation (23) is written as follows: is written as follows: 2Dp (r ) 1 2 LDR(r ) 2Δp (r ) t,is b 1 2 LDR(r ) b * t,is b f (r ) = St (24) PVS f (r ) = St (24) PVS b r c C (r ) N n K C (r ) c a L L ρ c C (b r ) N n K C (r b ) c a L b L b where N is the blade count and n is the rotor speed. The right-hand side of Equation (24) where N is the blade count and n is the rotor speed. The right-hand side of Equation (24) can be rearranged according to Equation (25) in the following way: In the first term, the can be rearranged according to Equation (25) in the following way: In the first term, the global aerodynamic performance characteristics are grouped. The second term represents global aerodynamic performance characteristics are grouped. The second term represents the operational condition of the rotor. In the third term, the basic parameters of the blade the operational condition of the rotor. In the third term, the basic parameters of the blade geometry are gathered. The fourth term represents the aerodynamics parameters of the geometry are gathered. The fourth term represents the aerodynamics parameters of the elemental blade section. Finally, the last term incorporates the empirics and coefficients. elemental blade section. Finally, the last term incorporates the empirics and coefficients. Dp (r ) 1 1 LDR(r ) 4 St Δp (r ) 1 1 LDR(r ) 4 St t,is b b t,is b f (fr ) (= r ) = (25) PVS b (25) PVS b 2 2 2 2 * r n K N c C (r ) aρ n N c C (r ) K a L L b In In order order t to o obt obtain ain gener generalized alized conc conclusions lusions a and nd tr trends, di ends, dimensionless mensionless q quantities uantities and and gr groups m oups must ust be be introduced. It should be introduced. It should be emphasized emphasized that PVS that PVS fr frequency is constant along equency is constant along the blade span due to the presented fan design method. Therefore, any characteristic taken the blade span due to the presented fan design method. Therefore, any characteristic taken at an arbitrary r radius can be replaced by the same characteristic taken at R . Further- at an arbitrary rb radius can be replaced by the same characteristic taken at Rmid. Further- b mid more, as a brief approximation h (r ) = h (R )  constant is presumed. With respect to more, as a brief approximation ηtt(rb) = ηt(tRmid) ≈ constant is presumed. With respect to the b mid the foregoing, the first term on the right-hand side of Equation (25) can be expressed: foregoing, the first term on the right-hand side of Equation (25) can be expressed: Δp (r ) Δp (R ) Δp (R ) Ψ (R ) t,is b t,is mid tip Dp (r ) Dp (R ) t mid t mid Dp (R ) Y (R ) tip t,is t,is t t b mid mid mid = = = (26) = = = (26) ρ ρ ρ η (R ) η 2 r a r a r h a ( tR mid ) t h 2 a a a t mid t where utip = Dtipπn is the tip circumferential speed. The second and the third terms are written as follows: Int. J. Turbomach. Propuls. Power 2022, 7, 2 16 of 23 where u = D n is the tip circumferential speed. The second and the third terms are tip tip written as follows: 1 1 1 1 1 1 1 = D  = s (27) tip tip 2 2 2 n c N n D  c N u c tip tip Thus Equation (25): s u s Y (R ) 1 LDR(R ) 4St Y (R ) LDR(R ) 2 St tip tip tip tip t mid mid t mid mid f (R ) = = (28) PVS mid 2 2  2 2 h 2 u c C (R ) K h (R ) c C (R ) K t tip t mid mid mid L L As it is outlined previously, the mechanically unfavorable design case is considered when the spanwise constant PVS frequency coincides with the first bending eigenfrequency of the fan blade, posing an increased risk of blade resonance: f = const = f (29) PVS B,1 If Equations (28) and (13) are substituted in the right-hand side and the left-hand side of Equation (29), respectively, the following equation is obtained: u s Y (R ) LDR(R ) 2 St 0.56 t tip tip mid mid = K t a (30) b b 2 2  2 h (R ) c C (R ) K l mid mid After rearrangement, Equation (30) can be written in the following as dimensionless, from: u C (R ) h (R ) t c 1 0.56 K tip t mid mid = K (31) a Y (R ) c s LDR(R ) 2St tip AR b mid mid Equation (31) provides a means for calculating the critical u /a velocity ratio for tip b 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 can be obtained (cf. Equation (10) and the paragraph below), and thus, the critical u value can be computed. Hence, critical tip rotor diameter X rotor speed data couples can be discovered for an entire rotor family, consisting of rotors of various diameters and speeds. If the rotor diameter and the nominal rotor speed are fixed for further defining a specific case study, the resultant, nominal u value can be compared to the aforementioned tip 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 , using Equation (25), for acoustics evaluation. PVS 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 away from this plateau in blade design, by selecting the appropriate operational and PVS geometrical characteristics, gives a potential for moderating the impact of fan noise on humans. If such design intent cannot be realized for modifying f , the PVS phenomenon PVS 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. Int. J. Turbomach. Propuls. Power 2022, 7, 2 17 of 23 5. Calculation Example for the Designed Rotor First, based on Equation (31) and data in Table 3, as it is specific to the PVS-affected rotor designed herein, the critical tip speeds are computed for various blade materials, which means different values of a in terms of the calculation process. The calculated values are summarized in Table 4. Table 4. Critical rotor tip speeds. Steel Aluminum Polycarbonate a [m/s] 5000 5100 1350 u [m/s] 1.80 1.83 0.49 tip,crit The methodology presented herein provides a means for simply checking whether PVS may cause a risk at all from a blade resonance point of view. Based on Table 4, it can be concluded that in the present case study, the critical tip speed is sufficiently low to make these cases irrelevant for the blade resonance point of view of the first-order bending mode (Figure 6). Namely, only impractically low rotor diameters D and/or rotor speeds n would result in coincidence of the f and f [Equation (10)] values for the presented PVS B,1 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. The reader is reminded that the mechanical effect of the blade twist is neglected herein, cf. Section 2. With consideration of the blade twist, the first bending eigenfrequency would be less, thus reducing the impractically low critical tip speed even further. As the second part of the calculation example, the rotor diameter and rotor speed are fixed as follows: D = 0.900 m, n = 1450 1/min. Such values are relevant in industrial tip ventilation. They result in u = 68.3 m/s. Considering data in Table 4 as well as previously tip 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. Table 5. PVS frequency for a five-bladed, n = 1450 1/min axial flow fan. D c C (R ) LDR (R ) s f tip L mid mid tip PVS 0.900 m 0.120 m 1.30 43.5 0.565 4520 Hz The calculated PVS frequency falls within the third-octave band of the middle fre- quency of 5 kHz. Thus, it approximates the plateau of the A-weighting graph. Therefore, the related noise may cause increased annoyance for a human observer. However, by mod- ifying the design parameters, the axial fan can be redesigned to keep the PVS frequency away from the plateau of the A-weighting graph, based on the presented computation. The systematic exploration of the advantageous parameter modifications, while finding reasonable compromises with other design perspectives, is also a future task. Based on Table 2, for the first bending eigenfrequency of the designed blade, the following value is calculated: f = 116 Hz. For the sake of completeness, the following B,1 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 fre- quency. The rotational speed n = 1450 1/min, which corresponds to 24 Hz, is thus  20% of the first bending eigenfrequency, which is satisfactory. On the other hand, as mentioned in Section 3.1, elements located upstream of the rotor may lead to rotor–stator interaction, occurring at a frequency of rotational frequency n Int. J. Turbomach. Propuls. Power 2022, 7, 2 18 of 23 multiplied by the number of upstream elements. Therefore, it is necessary to examine for which number of upstream elements (e.g., support strut, inlet guide vane) the computed eigenfrequency would coincide with the excitation frequency of the rotor–stator interaction. As f  5  24 Hz = 120 Hz, the critical element number is five, which should be avoided B,1 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 Based on the semi-empirical model in literature, the pessimistic design condition of spanwise constant PVS frequency is determined from an aerodynamic approach. To fulfill this condition, an iterative fan design method was elaborated, resulting in a design case study of a PVS-affected rotor. The frequency of PVS was computed with the knowledge of the global operational and geometrical characteristics of an axial fan. A calculation process for determining the eigenfrequency related to the first bending mode of vibration of a circular-arc-cambered plate blade was presented. By combining these two approaches, guidelines can be formulated, already in the preliminary design phase, for the following. (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. As a future objective, opposing the pessimistic design scenarios, redesign efforts are to be made for moderating/avoiding blade resonance and/or noise annoyance. Such efforts incorporate actions against the constancy of PVS frequency along the span. Systematic redesign scenarios as well as comparative experiments—also incorporating pessimistic, PVS-affected rotor cases—will be realized in the future for validating the methodology presented herein. The design aspects related to PVS are to be investigated in the future via studies on 2D—i.e., rectilinear—blade models as well as on truly 3D rotor blade geometries— unavoidable for full consideration of realistic rotor flow effects—by the concerted means of computational fluid dynamics (CFD), analytical mechanics, finite-element mechanical computations, and experimentation. Such studies will also serve as the exploration of the effects of assumptions and simplifications made in the method presented herein. Author Contributions: G.D. and J.V. conceptualization, methodology, investigation, and writing. All authors have read and agreed to the published version of the manuscript. Funding: This work has been supported by the Hungarian National Research, Development and Innovation (NRDI) Centre under contract No. NKFI K 129023. The research reported in this paper and carried out at BME has been supported by the NRDI Fund (TKP2021, project no. BME-NVA-02, TKP2021-EGA, and TKP2020 NC, Grant No. BME-NCS) based on the charter of bolster issued by the NRDI Office under the auspices of the Ministry for Innovation and Technology of Hungary. The contribution of Gábor DAKU has been supported by Gedeon Richter Talent Foundation (registered office: 1103 Budapest, Gyömroi ˝ út 19-21.), established by Gedeon Richter Plc., within the framework of the “Gedeon Richter PhD Scholarship”. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Acknowledgments: The authors are thankful to Tamás KALMÁR-NAGY for his assistance in the blade mechanics analysis. Conflicts of Interest: The authors declare no conflict of interest. Int. J. Turbomach. Propuls. Power 2022, 7, 2 19 of 23 Abbreviations The following abbreviations are used in this manuscript: Latin Letters A Cross-section [m ] 1/2 a Wave propagation speed = (E/r ) [m/s] b b AR Aspect ratio = l/c [-] b Transversal dist. between vortex rows [m] c/s Blade solidity [-] D Rotor diameter [m] d Trailing edge thickness [m] TE 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 ] I Polar moment of area [m ] I Torsional constant [m ] K* Empirical coefficient  1.2 [-] K Preliminary design constant [-] K Blade mechanics coefficient [-] K Mechanical constant [-] K Bending mode constant [-] B,i c Blade chord length [m] c Axial velocity component [m/s] C Drag coefficient [-] C Lift coefficient [-] l Blade span [m] LDR Lift-to-drag ratio = C /C [-] L D N Blade count [-] n Rotor speed [1/s] R Dimensionless radius = r/r [-] tip r Radial coordinate [m] Re Reynolds number = c U /n [-] c a s Blade spacing = 2r/N [m] St Strouhal number [-] St* Universal Strouhal number [-] t Profile thickness [m] t/c Relative thickness [-] u Rotor circumferential velocity [m/s] U Free-stream velocity = (w + w )/2 [m/s] 0 1 2 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] Int. J. Turbomach. Propuls. Power 2022, 7, 2 20 of 23 Greek Letters a Angle of attack [ ] b l Vibration constant  (2i 1)/2 g Stagger angle [ ] d Specific diameter [-] z Rotation angle [rad] Dc Increase of tangential velocity [m/s] Dp Isentropic total pressure rise [Pa] t,is h Efficiency [-] Q Momentum thickness of blade wake [m] n Hub-to-tip ratio [-] n Kinematic viscosity of air [m /s] n Poisson’s ratio [-] r Density [kg/m ] s Specific speed [-] t Time [s] j Local flow coefficient [-] F Global flow coefficient [-] y Local pressure rise coefficient [-] Y 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 Int. 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Journal

International Journal of Turbomachinery, Propulsion and PowerMultidisciplinary Digital Publishing Institute

Published: Jan 7, 2022

Keywords: axial flow fan; blade vibration; low tip speed; preliminary fan design; vortex shedding

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