# REGENERATION OF DUCTED ROTOR-STATOR WAKE-INTERACTION TONAL NOISE BECAUSE OF VANE-TO-VANE IRREGULARITIES

Michel Roger, A. Pereira

► To cite this version:

Michel Roger, A. Pereira. REGENERATION OF DUCTED ROTOR-STATOR WAKE-INTERACTION TONAL NOISE BECAUSE OF VANE-TO-VANE IRREGULARITIES. 29th International Congress on Sound and Vibration (ICSV29), International Institute of Acoustics and Vibration (IIAV), Jul 2023, Prague, Czech Republic. hal-05328677

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# REGENERATION OF DUCTED ROTOR-STATOR WAKE-INTERACTION TONAL NOISE BECAUSE OF VANE-TO-VANE IRREGULARITIES

Michel Roger and Antonio Pereira

École Centrale de Lyon, France

email: michel.roger@ec-lyon.fr

This paper theoretically investigates the regeneration of tonal noise in a ducted rotor-stator configuration, for which the blade/vane counts should ensure a complete cut-off of the first two blade-passing frequency tones. The re-emergence of the tones results from the modification of some of the vanes, needed to meet structural constraints. The associated deviations from pure vane-to-vane periodicity de-activate the necessary destructive interference for cut-off. They cause a substantial change of the modal structure of the in-duct sound field. Simple analytical predictions, accessible at the early design stage, are proposed to assess this modal structure from a minimum description of the modification. They are applied to a true, subsonic, axial-flow rotor-stator fan architecture, typical of aircraft air-conditioning systems, for which measurements were made on a dedicated test bench. Two versions of the stator, either with identical vanes or with some thickened ones, were tested. The measured trends are found in good qualitative agreement with the analytical predictions. The latter are confirmed as a reliable low-noise design tool.

Keywords: aeroacoustics, fan noise, analytical modeling, in-duct propagation

## 1. Introduction

When designed with identical and evenly spaced blades and vanes, a ducted rotor-stator stage generates sound at multiples of the Blade-Passing Frequency (BPF), defined as the rotational frequency $\Omega/(2\pi)$ multiplied by the number of blades $B$, because of periodic aerodynamic interactions. Noise is attributed to the impingement of rotor wakes on the stator vanes (or Outlet Guide Vanes, OGV), and to the contamination of rotating blades by the potential flow distortion around the vanes. The sources are the blade and vane periodic forces, acting as dipoles, as stated by the acoustic analogy [1, 2]. Both mechanisms are referred to as wake-interaction noise and potential-interaction noise, respectively. They have the same modal structure, in the sense that they excite the same acoustic duct modes, of azimuthal and radial orders $m$ and $j$. A mode is transmitted along the duct, and possibly radiates outside, only if the operating frequency exceeds its cut-off frequency; otherwise, it is said cut-off. The angular cut-off frequency is $\omega_c(n, j) = c_0\beta_a K_{nj}$, where $n = |m|$, $c_0$ is the sound speed and $\beta_a = \sqrt{1 - M_a^2}$, $M_a = W_0/c_0$ being the Mach number of the axial mean flow, assumed uniform. The values $K_{nj}$ are increasing functions of the indices $n$ and $j$, defined by the outer duct radius $R_2$ and the hub-to-tip ratio $\sigma = R_1/R_2$. A classical noise-reduction mean is to tune the blade and vane counts, $B$ and $V$, in such a way to ensure the cut-off
of all spinning duct modes of the tonal aerodynamic noise, at the first blade-passing frequencies. It is applied, for instance, on subsonic fans dedicated to avionics cooling and aircraft air-conditioning systems, characterized by moderate Mach numbers. However, some of the stator vanes may need to be modified, for instance to comply with structural constraints during the manufacturing process Such a situation is addressed in the present work, in the continuation of previous studies [8, 7]. Configurations with identical vanes and with some modified vanes are said homogeneous and heterogeneous, respectively.

In heterogeneous systems, cut-on duct modes are regenerated because of the deviations from vane-to-vane regularity, making the a priori low-noise design, based on blade/vane counts, inoperative. Keeping the regenerated sound at an acceptable level is a crucial need. For this, simple prediction tools, usable at the early design stage and able to produce relevant estimates, are needed. The present work describes a simple approach, aimed at quantifying the regenerated sound for pre-identified vane modifications, with a minimum computational effort. The approach relies on an in-duct analytical model of the sound radiated by a rotor-stator stage. It only needs a very rough description of the system, including blade and vane numbers, mean flow rate, dimensions and rotational speeds. Details, such as the exact shapes of blade and vane cross-sections, are ignored. Advanced numerical means, much more computationally demanding, would ensure more accurate predictions, but they would also require a complete definition of the system, often unavailable at the early design stage.

The mathematical background is reminded in section 2. The methodology is assessed in section 3 by comparing the predictions with measured tone levels on a dedicated axial-flow fan test-bench. The fan architecture includes a stator with three thickened vanes, enabling the mounting on the duct wall. Another version with a homogeneous stator row, specially manufactured for the study, has also been tested as a reference. Finally, remarks and conclusions are given.

# 2. Theoretical Background

## 2.1 In-Duct Formulation

In order to remain compatible with closed-form expressions of the sound field, a simplified in-duct formulation is accepted here, assuming uniform axial mean-flow and annular duct-shape of constant cross-section. For this, the hub-to-tip ratio at the stator location is considered, since the sources of the wake-interaction noise are distributed on the stator vanes. The hard-walled, annular-duct Green's function for the convected Helmholtz equation, given for instance in [2], is used as a background. For a point dipole of strength $F(\mathbf{r}_0) = (0, F_\phi, F_z)$ at point $\mathbf{r}_0 = (r_0, \phi_0, z_0)$, thus assuming a zero radial contribution, the sound pressure at angular frequency $\omega = \mu B \Omega$, in cylindrical coordinates $(r, \phi, z)$ with $z$ aligned in the axial flow direction, and for the case of upstream propagation, is expressed as $p_{\mu B}(\mathbf{r}, t) = p_{\mu B}(\mathbf{r}) e^{-i\mu B \Omega t}$, with

$$p_{\omega}(\mathbf{r}) = \frac{-1}{2} \sum_{m=-\infty}^{\infty} \sum_{j=0}^{\infty} \frac{f_{nj}(r) f_{nj}(r_0)}{\Gamma_{nj} \sqrt{k^2 - \beta_a^2 K_{nj}^2}} e^{im(\phi-\phi_0)} e^{-i\gamma_{nj}(z-z_0)} \left(F_z \gamma_{nj} - \frac{m}{r_0} F_\phi\right), \quad (1)$$

with $n = |m|$, $k$ being the acoustic wavenumber, $f_{nj}(r) = J_m(K_{nj}R_2) + X_{nj} Y_m(K_{nj}R_2)$ and

$$\gamma_{nj} = \frac{1}{\beta_a^2} \left[ M_a k + \sqrt{k^2 - \beta_a^2 K_{nj}^2} \right], \quad \Gamma_{nj} = \int_0^{2\pi} \int_{R_1}^{R_2} f_{nj}^2(r) r dr d\phi .$$

The eigenvalues $K_{nj}$, $X_{nj}$ are determined by solving the equations of rigid-wall boundary conditions [2].

The in-duct sound field of an OGV row is obtained from this generic formula by linear superposition principle. For this, the unsteady lift forces induced on a vane and acting as distributed dipoles
are weighted by phase factors and integrated over the vane surface. The phase factors account for propagation-path differences. Furthermore, for a homogeneous stator row, all vanes have the same contribution, except for a phase shift. The latter is defined by the time delay between the impingements of a rotor-blade wake onto successive vanes, responsible for azimuthal interference. Practically, a radial discretization into annular strips is performed from hub to tip, considering locally homogeneous flow conditions in each strip of mean radius $r_\ell$ and span $\Delta r$. The chordwise integration is derived analytically, resorting to the compressible classical unsteady aerodynamic theory shortly described below. This framework ensures that non-compatness effects of the OGV are properly included. It leads to the calculation of generalized force coefficients, in place of the factors $F_z$ and $F_\phi$. The following expression is obtained:

$$p_{\mu B}(\mathbf{r}) = \frac{-V}{2} \sum_{s=-\infty}^{\infty} \sum_{j=0}^{\infty} \frac{f_{nj}(r)}{\Gamma_{nj} \sqrt{k_{\mu B}^2 - \beta_a^2 K_{nj}^2}} e^{i(\mu B - sV)\phi} e^{-i\gamma_{nj}^{(\mu B)}(z-z_0)} T_{sj}, \quad (2)$$

with $n = |\mu B - sV|$ and

$$\gamma_{nj}^{(\mu B)} = \frac{1}{\beta_a^2} \left[ M_a k_{\mu B} + \sqrt{k_{\mu B}^2 - \beta_a^2 K_{nj}^2} \right], \quad k_{\mu B} = \frac{\mu B \Omega}{c_0},$$

$$T_{sj} = \sum_{\ell=1}^{N} f_{nj}(r_\ell) \left\{ \gamma_{nj}^{\mu B} \cos \gamma_S - \frac{\mu B - sV}{r_\ell} \sin \gamma_S \right\} F_{sj}(r_\ell).$$

A reference vane ($j=0$) is assumed at zero azimuth $\phi_0$ without loss of generality, in the cylindrical coordinates $(r, \phi, z)$ associated with the axes $(e_x, e_y, e_z)$ in Fig. 1-a. In Eq. (2), each term of the sum, weighted by a Bessel function, features a spinning mode of in-duct transmission, of azimuthal and radial orders $m$ and $j$, with the angular phase speed $\Omega_s = \mu B \Omega / (\mu B - sV)$ and a number $m = \mu B - sV$ of angular periods, called lobes. The selection of the excited spinning mode orders is known as Tyler & Sofrin's rule [10]. It can be shown that only the cut-on modes, for real values of $\gamma_{nj}^{\mu B}$, carry energy [2]. Therefore, the implemented quantity in the following is the acoustic power defined by intensity integration over an inlet-cross-section, involving the sum on all cut-on selected mode orders $(m, j)$.

<figure><img src="image_3.png" /><figcaption>Figure 1: Sets of coordinates for the implementation of the stator-noise model. (a): general view; note that the OGV are untwisted. (b): unwrapped strip at hub radius, also showing blade-tip cross-section and tangential shift of the impingement $\Delta_W$ between hub and tip, due to blade twist.</figcaption></figure>
In the present work, the unsteady loads on the vanes are estimated with Amiet's theory. For this, a vane is assimilated to a flat plate and its response to incident velocity disturbances is derived in Cartesian coordinates, unwrapping the cylindrical cut of radius $r_\ell$, as depicted in Fig. 1-b. The cylindrical coordinates of a source point of coordinate $y_1$ along the vane chord, from $-c/2$ at leading edge to $+c/2$ at trailing edge, are $\phi_0 = -y_1 \cos \gamma_S / r_0$ and $z_0 = y_1 \sin \gamma_S$. Derivations are not detailed here; some background is provided in reference [4]. Finally, the force factor in a strip is found as $F_{sj}(r_\ell) = \Delta r (\mathcal{F}_1 + \mathcal{F}_2)$, with

$$ \mathcal{F}_1 = \frac{-4 \rho_0 U_0 w_\mu e^{-i \Theta_2}}{\sqrt{(1 + M_0) k_1^*}} \frac{c}{2} \frac{E(2 \Theta_1)}{\sqrt{2 \Theta_1}}, $$

$$ \mathcal{F}_2 = \frac{-2 \rho_0 U_0 w_\mu e^{-i \Theta_2}}{\sqrt{2\pi (1 + M_0) k_1^* \Theta_1}} \frac{c}{2} \left\{ i (e^{2i\Theta_1} - 1) + (1 + i) \left[ E(4\bar{\mu}) - 2\sqrt{\bar{\mu}} e^{2i\Theta_1} \frac{E^*(2\Theta_3)}{\sqrt{2\Theta_3}} \right] \right\}, $$

$$ k_1^* = \frac{\mu B \Omega c}{2U_0}, \quad \bar{\mu} = \frac{k_1^* M_0}{\beta^2}, \quad \Theta_2 = \frac{c}{2} \left\{ \sin \gamma_S \gamma_{nj} + \cos \gamma_S \frac{m}{r_\ell} \right\}, $$

$$ \Theta_1 = \Theta_2 + (1 - M_0) \bar{\mu}, \quad \Theta_3 = \Theta_2 - (1 + M_0) \bar{\mu}, \quad E(\xi) = \int_0^\xi \frac{e^{it}}{\sqrt{2\pi t}} dt, $$

$$ \beta = \sqrt{1 - M_0^2}, \quad M_0 = \frac{U_0}{c_0}, \quad U_0 = \frac{\tan \gamma_R \Omega r_\ell}{\sin \gamma_S + \cos \gamma_S \tan \gamma_R}. $$

In these expressions, $\gamma_S$ and $\gamma_R$ are the angles of the vanes and of the wakes with respect to a cross-section of the duct, both defined as positive (Fig.1-b). $U_0$ is the phase speed of the wake-vane interaction, used as characteristic speed in Amiet's formulation. $w_\mu$ denotes the complex amplitude of the upwash experienced by a vane at the considered frequency. This parameter is the Fourier coefficient of the periodic velocity deficit in the wake at the location of the stator leading edge. Its amplitude and decrease with $\mu$, as a function of the rotor-stator separation $d_{RS}$ and of the wake angle $\gamma_R$, must be provided. The present application relies on an empirical wake model proposed by Majjigi & Gliebe [3]. The model expression for the ratio of centerline to external velocities is

$$ \frac{1 - w_0}{U_e} = C_D^{1/4} \frac{1.95 + 0.3675 (x/c_R)}{1 + 7.65 (x/c_R)}, $$

where $U_e$ is assimilated to $\Omega r_\ell / \cos \gamma_R$. $c_R$ and $x$ stand for the rotor blade chord and the distance from the trailing edge along the wake, respectively. The drag coefficient $C_D$ has been arbitrarily taken as 0.01. In the present application, the radial split into strips has been made with five vane segments. It is worth noting that, though physically consistent, all assumptions make the modeling approach approximate. For this reason, the predictions must be taken with care, more dedicated to the assessment of relative variations than able to produce accurate absolute levels.

## 2.2 Vane-Response Variations

The present work addresses heterogeneous responses of equally spaced vanes. The response of a stator with vane-to-vane differences is assumed the sum of two contributions. The first one is an ideal homogeneous-stator response, corresponding to Eq. (2) and assuming the same harmonic loads on all vanes. The second one is an additional sound field, noted $p_{\mu B}^{(e)}(\mathbf{r})$, produced by error dipoles. The latter are defined by the differences of dipole strengths associated with modified and baseline (un-modified) vanes. The loading harmonic on the vane of index $j$ in the heterogeneous configuration is written as $F_{\mu B,j} = \xi_\mu F_{\mu B}$, where $\xi_\mu$ is a frequency-dependent factor, equal to 1 for baseline vanes. The error dipoles are
considered as the signature of another fictitious stator, with harmonic loads of strength $(1 - \xi_{\mu}) F_{mB}$, only made of the modified vanes. The expression of $p_{\mu B}^{(e)}(r)$ for the fictitious stator contribution is obtained from Eq. (1) by the same procedure as before, the sum being performed only on the indices $j = j_e$ of the modified vanes, with the additional factor $1 - \xi_{\mu}$. The factor $V$ must be replaced by the interference sum

$$ (1 - \xi_{\mu}) \sum_{\lambda=0,8,15} e^{i\lambda(\mu B - m)2\pi/V} $$

calculated on the three error dipoles only. Apart from the factors $\xi_{\mu}$ accounting for the response reduction of thicker vanes, this interference sum takes non-zero values for any mode order $m$, making Tyler & Sofrin's rule meaningless in its usually accepted form.

This splitting of the sound field allows to identify the mode orders $(m, j)$ that are regenerated by the response heterogeneity. The factors $\xi_{\mu}$ still need to be estimated to predict the amplitude of these modes. They depend on the nature of the modification. The latter can be an increased chord length, as investigated by Roger & Caule [8], or the presently addressed increased thickness of some vanes at constant chord, detailed in section 3.1. In such a case, previous works dealing with the noise from impinging turbulence on isolated airfoils show evidence of a reduced response, due to leading-edge thickness and rounding, compared to thin airfoils (see Roger & Moreau [9] for a short review). The noise reduction in decibels $\Delta_{dB}$ is proportional to frequency and to relative thickness, according to an empirical fit involving the integral length scale of the turbulence and the chord length of the airfoil, not detailed here for conciseness. This fit has been retained for a first insight into the effect of modified vanes, using the estimated half-wake width at the location of the interaction with the leading edges of the vanes in place of the integral length scale. In absence of more reliable information, the response of the thicker vanes is finally expected about 60-70% of that of the baseline vanes. Though a more precise assessment would require dedicated studies, the formulation is believed to provide relevant orders of magnitude of the expected sound regeneration.

# 3. Comparison with Experiment

## 3.1 Low-Speed Fan Test-Case

<figure><img src="image_4.png" /><figcaption>Figure 2: (a): test fan architecture, thicker vanes featured in red, and stator reference frame. (b): typical sound spectrum in the inlet duct at 10,000 rpm, averaged over a set of microphones. Homogeneous stator (black) versus heterogeneous stator (red).</figcaption></figure>

A low-speed axial-flow fan test bench of Ecole Centrale de Lyon is used in this section to assess the methodology. The test bench is aimed at characterizing the effect of structural modifications. It is equipped
with a turbulence control screen suppressing inlet distortions. The fan is a rotor-stator stage with $B = 17$ blades and $V = 23$ outlet guide vanes. The blade-tip radius is 85 mm and the hub-to-tip ratio is 0.55. The inner duct wall is equipped with flush-mounted microphones for in-duct sound measurements. This instrumentation allows to perform a modal breakdown of the sound field, both upstream and downstream of the stage [6]. The experiment and key results are described in more details in an accompanying paper [5]. The rotation speed is 10,000 rpm, corresponding to a tangential tip Mach number of 0.26. This definition would ensure the cut-off of the first two BPF tones, in the case of homogeneous blade and vane rows, based on Tyler & Sofrin's rule. But acoustic measurements, performed both outside and inside the duct, evidenced a strong emergence of these tones. This was attributed to the fact that three from the 23 vanes, at equivalent vane positions $j_e 2\pi/23$, with $j_e = 0, 8$ and $15$, are manufactured about three times thicker, to receive screws for fan fixing on the outer duct wall (see Fig. 2-a). Numerical simulations achieved with a Lattice-Boltzmann Method (LBM) suggested that, in view of the rotor-stator distance, potential-interaction noise is much weaker than wake-interaction noise [7]. The regenerated sound *a priori* results from a different response of the thickened vanes to the incident wakes. The present investigation is aimed at confirming this conclusion, and at showing that it could be foreseen with simple analytical arguments.

## 3.2 Main Results

Sound spectra measured inside the duct and averaged over all microphones of the inlet are plotted in Fig. 2-b, for both homogeneous (black) and heterogeneous (red) versions of the stator. With the latter, the sound level is increased by about 9 dB at the BPF and 3 dB at twice the BPF. The remaining residual sound level at the first two BPF tones of the homogeneous configuration suggests that other deviations from ideal flow contribute, such as localized separation areas. This point still needs to be confirmed. The third BPF tone is found almost unchanged; the modification has a weak effect on it, because it is already cut-on in the homogeneous configuration. Furthermore, no significant change is seen on the broadband noise, the sources of which are uncorrelated and excite all possible duct modes.

<figure><img src="image_5.png" /><figcaption>Figure 3: Predicted modal powers of the fictitious stator (empty bars) at the first three BPF tones, compared to measurements (blue bars). Homogeneous-stator prediction added at 3BPF. Some of the dominant regenerated modes pointed by arrows, for discussion. High-and-low predicted levels correspond to limit values of the unknown factors $\xi_\mu$.</figcaption></figure>

Sound power breakdowns of the tonal noise on the azimuthal mode orders in the inlet duct, all radial modes included, as deduced from predictions and measurements, are compared in Fig. 3. Remind that
for the first two BPF tones, zero sound power is expected in the homogeneous configuration. Therefore, experimental results are shown for the heterogeneous configuration and predictions correspond to the fictitious stator. Predicted levels for the regenerated modes are globally underestimated. This is attributed to the lack of precise information about the input data in the models, of crucial effect in the present case because of the non-compactness of the OGV row. But the overall shapes of the bargraph envelopes are in good qualitative agreement. In particular, most of the dominant regenerated modes are reproduced by the model. At the BPF, stator heterogeneity dominantly regenerates the modes +3 and -3, and less crucially the symmetric mode $m = 0$. The best qualitative agreement between the predictions and the measurements is found at this frequency.

At 2BPF, sound is mostly regenerated on the modes -6, -3, 0, 2, 3 and 5 according to the predictions, whereas all cut-on modes are observed in the experiment with similar levels, except for the emerging mode $m = 5$. At 3BPF, many cut-on modes are regenerated with the heterogeneous stator, whereas only the mode $m = 5 = 3B - 2V$ is selected by Tyler & Sofrin's rule in the homogeneous configuration. The strongly emerging modes 2, 5, 8 and 11, indicated by arrows in the figure, are found both in the measurements and in the predictions. Therefore, they are believed to have true features of the effect of heterogeneity. However, the sound from the error dipoles remains about 25 dB below the level of the mode $m = 5$ characteristic of the homogeneous stator. This confirms that heterogeneity has a very weak effect at this frequency, as already seen in the sound spectrum plotted in Fig. 2-b.

## 4. Conclusions

A combination of simple analytical models of sound generation and in-duct propagation is proposed, to predict the tonal rotor-stator wake-interaction noise of a rotor-stator stage, based on a minimum description of the stage. The approach is aimed at inferring the possible regeneration of forbidden modes, when some of the stator vanes are modified for structural reasons, in a preliminary design context. Assumptions are needed for the implementation, about missing or not yet specified details, such as vane response deviation due to the modification, vane cross-section shape, rotor wake diffusion, and so on. This is why empirical/analytical wake and response models are used to produce the input data. Despite the associated uncertainties, the approach is physically consistent. Blind predictions have been made and assessed on the case of a dedicated fan test bench, with homogeneous and heterogeneous versions of the stator. The first two BPF tones are theoretically cut-off with the former and regenerated with the latter. The comparison led to a partial agreement. The same dominant regenerated modes were found in the predictions and in the measurements at the BPF and 3BPF. This validates the analytical approach as a fast tool, for both diagnosis of existing systems and preliminary design. The poorer agreement found at 2BPF might be the effect of a more critical frequency for the model assumptions. It is also attributable to another source of tonal noise in the experiment, as suggested by the significant residual levels observed at the first two BPF tones in the homogeneous configuration.

The main outcome is that even minor local modifications on a stator row are able to regenerate high levels of tonal noise, compared to a homogeneous configuration with perfectly identical vanes. Heterogeneity in stator vane responses introduces imbalance in distributed coherent sources, which would interfere destructively otherwise to ensure cut-off. Therefore, it has a minor effect on tones that are already cut-on in homogeneous configuration, and a negligible effect on broadband noise sources.

## Acknowledgements

This work was performed within the framework of the LABEX CeLyA (ANR-10-LABX-0060) of Université de Lyon, within the program Investissements d'Avenir (ANR-11-IDEX-0007) operated by the
French National Research Agency (ANR).

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