Numerical assessment of the tonal noise of Counter-Rotating Open Rotors at approach

Abstract

A hybrid method combining a three-dimensional unsteady Reynolds-Averaged Navier–Stokes simulation of a cropped Counter-Rotating Open Rotors at approach conditions and an acoustic analogy either based on a time formulation, an advanced-time formulation of Ffowcs-Williams and Hawkings’ analogy, or on a frequency formulation, an extension of Hanson’s model to non compact chord length, has shown three main sources, the impacts of the front-rotor wakes on the aft-rotor, of the front-rotor tip vortex on the aft-rotor tip, and of the front-rotor hub horse-shoe vortex on the aft-rotor blade foot. Consequently, this study confirms that the aft-rotor is the dominant tonal noise source and has identified another potential noise source when the Counter-Rotating Open Rotors is installed caused by the strong tip vortex of the highly loaded cropped aft-rotor. The present rotor–rotor distance should not be reduced as the potential effect from the aft rotor is already felt on the front rotor. The influence of several numerical parameters including grid refinement, data sampling, and simulation length has been evaluated on both aerodynamic and acoustic performances.

Keywords

Counter Rotating Open Rotor Propeller noise acoustic analogy

Introduction

Counter-Rotating Open Rotors (CROR) in the low to medium range transportation are seen as good propulsion systems to significantly improve the specific fuel consumption of the next generations of aircrafts.¹ Nevertheless, their acoustic design is a critical issue,² and has been a major obstacle so far for its installation on aircrafts. Indeed, the aerodynamic interactions between the two rotors are quite strong as shown by Podboy and Krupar,³ generating an significant noise over a very large spectrum and in all directions with a high tonality, contrarily to the classical turbofans where the duct around the fan limits the noise propagation to some frequencies and directions. Moreover, dedicated liners in the nacelle can reduce the remaining tonal content at the engine inlet and outlet. The rotor interactions in a CROR can be split into two distinct components. The first one is the most annoying tonal emission produced by the periodic interactions of deterministic flow structures on each rotor. Tonal sources not only include the impingement of vortical structures such as wakes or tip vortices coming from the front rotor onto the downstream rotor blades but also the surface-pressure fluctuations induced on the front rotor by the potential effects coming from the aft rotor. The second acoustic contribution is the broadband noise generated by the interaction of the random turbulent structures in rotor wakes or in the incoming flow with the blades. In a CROR, the tonal noise is often the most annoying feature because of the loud tones dominating a lower level broadband noise. The precise understanding of the tonal noise generation and propagation is therefore of prime importance to design future CRORs. This is the focus of the present study in the particular approach flight conditions.

A first objective is to assess the numerical settings and simulation temporal characteristics to predict both aerodynamic performances and aeroacoustic tonal sources accurately. A second goal is to compare two different noise models based on an acoustic analogy in free-field, a modified Hansonʼs model in the frequency domain and an advanced-time formulation of the classical Ffowcs-Williams and Hawkingsʼ analogy in the time domain. A third objective is also to check the noise sources at another critical operating condition (approach) rather than the usual cruise configuration that the CROR has been designed for. It also provides a complementary configuration to the take-off situation recently chosen by Colin et al.^4–6 The computational approach including the CROR configuration, the hybrid method combining unsteady aerodynamic simulations and acoustic analogies are first described in “Computational approach” section. The aerodynamical and acoustical results are then shown in the “Results” section.

Computational approach

CROR configuration

The unsteady flow field for a realistic isolated modern CROR configuration (hub and blade geometries) has been recently achieved to mimic tests performed at NASA.^7,8 The latter were performed on a scaled model in the 9′ × 15′ Low Speed Wind Tunnel using the refurbished Open Rotor Propulsion Rig on an improved evolution of the historical F31/A31 CROR configuration that has been recently extensively revisited both experimentally and numerically.^9–11 The latter does not take into account installation effects such as the pylon supporting the engine or the vicinity of the aircraft body or wings. Moreover, the ambient flow is supposed uniform without general incidence relatively to the engine to reproduce the wind tunnel test setup. The simulated configuration (a single blade channel with identical blade number for both rotors has been duplicated here) is represented in Figure 1, along with the computational domain (transparent surfaces). The aft rotor is clipped (i.e. its diameter is reduced) to reduce the interaction with the wake of the front rotor in cruise conditions. The hub to tip ratio of the row is approximately 2. The front rotor diameter is approximately 4.2 m, the aft rotor being 10% smaller. The distance between the two rotors is approximately 2c, c = 0.4 m being the front rotor chord length. The two rotors have $B_{1} = 12$ and $B_{2} = B_{1} - 2$ blades, respectively.

Figure 1.

CROR configuration.

The study has been performed at approach conditions. The advancing Mach number is about M_x = 0.2 and the tip relative Mach number for the front rotor is subsonic (approximatively M₁ = 0.6). For the present design, both rotors rotate at the same absolute speed but in opposite direction ( $Ω_{1} = - Ω_{2} = 827 tr / \min$ ), with a quasi-equal low torque split corresponding to approach conditions (contrary to the high torque at take-off). The wind-tunnel pressure and temperature are set to P_s = 101325 Pa, T_s = 303.15 K, respectively. In the full experimental test matrix, these particular flow conditions were chosen because of the importance of noise emission during the take-off and landing phases. Note that an additional result of the present study is also to verify if the rotor clipping is enough in these new flow conditions.

Method for flow simulation

Since the main objective here is to analyze the sources for tonal noise, it is not necessary to achieve a very high degree of accuracy in the prediction of the small vortical structures associated with turbulence. As a consequence, Unsteady Reynolds-Averaged Navier–Stokes (URANS) have been chosen to capture the deterministic features of the flow field at a reasonable numerical cost. The computational domain is 20c long upstream of the front rotor, 15c long downstream of the aft one, and the outer diameter is approximatively 4 times bigger than the front rotor diameter D. A single blade passage is computed for each rotor, with a common azimuthal pitch corresponding to the front rotor one as explained below. It was also carefully checked that these dimensions had not effect on the flow performances and the vena contracta at approach.⁷

URANS simulations have been performed using the cell-centered structured solver Turb’Flow.¹² Spatial discretization was performed using the second-order upwind AUSM+-up scheme¹³ with the Van Albada limiter for the conservatives variables (density, momentum, and energy), whereas a first-order upwind scheme was used for the turbulent flow-field. Kokʼs two equation k-ω model¹⁴ is used for turbulence coupled with Menterʼs turbulence kinetic energy production limiter.¹⁵ Time discretization is achieved using an explicit second-order five-steps Runge-Kutta scheme yielding a time step of 3.2 $10^{- 8}$ s. Using a mesh fine enough to reach a maximum first cell dimensionless height $y^{+} = 5$ mandatory for the above turbulence model, there is about 5 million mesh points for the whole case (Figure 2). Soulat et al. showed that, for approach conditions (lower Mach number and Reynolds number based on the rotor chords), this mesh refinement was enough to yield global performances (thrust, torque, and efficiency) and flow features independent of the grid.⁷ Note that Colin et al. found for a similar but different CROR configuration that, at take-off (higher Reynolds number), at least 30 millions nodes were needed for resolved chimera and chorochronic URANS simulations.^4,5,6 Only on the high-frequency interaction tones ( $3 {BPF}_{1} + 3 {BPF}_{2}$ ), an even finer grid of 30 millions nodes was needed and no grid convergence was even achieved.⁵

Figure 2.

Meridional view of the fine mesh.

As the mesh does not physically move during the computation, the rotational effects (Coriolis and centrifugal forces) are directly taken into account as source terms in the URANS equations. The interface between the two rotors is achieved through a discrete Fourier decomposition and reconstruction of the flow field in the transverse direction; the relative movement of the rotors is simulated using a time variable offset in the flow-field reconstruction across the interface. This allows a complete unsteady communication between the interface blocks of each rotor. The other numerical boundaries bear either a no-slip wall conditions (blades, the rotating hubs, and the stationary nacelle in the appropriate reference frame), a slip condition (far-field boundary), periodicity, or inlet/outlet conditions. Momentum and density are imposed at the inlet, whereas static equilibrium is set at the outlet. A 5% turbulence intensity is also imposed at the inlet. The inner cylindrical surfaces protruding before and after the physical hub are only numerical interfaces and as such are treated with slip condition to avoid any boundary layer development before or after the stationary nacelle. The latter extends up to 0.27 m upstream of the front rotor and 0.2 m downstream of the second rotor. The hub of the front rotor then rotates in the clockwise direction up to approximately the middle of the inter-rotor space, and finally the hub of the aft rotor rotates in the counterclockwise direction up to the connection with the stationary nacelle. Note that, since the present solver does not perform chorochronic simulations, the number of blade on each rotor was set equal by adding two blades on the aft rotor to allow a periodic simulation of one blade on each rotor. This choice has been made to preserve the correct wake development of the front rotor, and consequently a proper excitation on the rear rotor. It was also checked that it had a negligible effect on the total thrust. The URANS is initialized with a converged RANS result, even though the latter yields a completely unphysical wake development from the front rotor as suggested below.

Method for acoustic computation

There are actually several possibilities to calculate the tonal acoustic emissions of a CROR, ranging from simple and fast analytical models to complete but very costly direct numerical simulations. Using an acoustic analogy, the different noise contributions of such a rotating machine can be split into three main contributions: (a) a monopole term caused by the fluid displacement by the blade, (b) a dipole term coming from the force exerted by the fluid on the blade, and (c) a quadrupole one originating in the fluid turbulence and possible volume sources. A preliminary study comparing the thickness noise and the steady and unsteady loading noise showed that, except for the first harmonic of the rear rotor in a narrow front arc, the thickness noise is negligible, and can be neglected for the present configuration. Similarly, the quadrupole noise is neglected since the Mach number is relatively low (much lower than 0.8). Consequently, only the dipole term is kept and the study is focused on the dominant wake interaction on the rear rotor.

Modified Hansonʼs model

The first technique that considers counter-rotating dipoles is the semi-analytical model proposed by Hanson.¹⁶ It predicts the far-field acoustic pressure $p (x, r, θ)$ for any CROR using the dipole source distribution over the blades. For the rear rotor (R₂), it reads in the time domain:

p_{R_{2}} (\vec{x}, t) = \frac{- i ρ a^{2} B_{2} \sin θ}{8 π (r_{1} / D) (1 - M_{x} \cos θ)} \sum_{m = - \infty}^{\infty} \sum_{k = - \infty}^{\infty} \exp (i ({mB}_{2} - {kB}_{1}) (φ - φ^{(2)} - π / 2)) \times \exp (i ({mB}_{2} Ω_{2} + {kB}_{1} Ω_{1}) (r / a - t)) \int_{hub}^{tip} M_{r}^{2} \exp (i (φ_{0} + φ_{s})) \times J_{{mB}_{2} - {kB}_{1}} [\frac{({mB}_{2} Ω_{2} + {kB}_{1} Ω_{1}) z_{0} (D / (2 a)) \sin θ}{1 - M_{x} \cos θ}] \times [k_{x} \frac{C_{Dk}}{2} Ψ_{Dk} (k_{x}) + k_{y} \frac{C_{Lk}}{2} Ψ_{Lk} (k_{x})] d z_{0}

(1)

where

k_{x} = \frac{D / a}{M_{r}} [\frac{{mB}_{2} Ω_{2} + {kB}_{1} Ω_{1}}{1 - M_{x} \cos θ} - {kB}_{1} (Ω_{2} + Ω_{1})] c / D

(2)

k_{y} = - \frac{2}{M_{r}} [\frac{({mB}_{2} Ω_{2} + {kB}_{1} Ω_{1}) M_{T} \frac{D}{2 a} z_{0} \cos θ}{1 - M_{x} \cos θ} - \frac{({mB}_{2} - {kB}_{1}) M_{x}}{z_{0}}] c / D

(3)

φ_{s} = \frac{D / a}{M_{r}} [\frac{{mB}_{2} Ω_{2} + {kB}_{1} Ω_{1}}{1 - M_{x} \cos θ} - {kB}_{1} (Ω_{2} + Ω_{1})] \frac{MCA}{D}

(4)

φ_{0} = \frac{2}{M_{r}} [\frac{({mB}_{2} Ω_{2} + {kB}_{1} Ω_{1}) M_{T} \frac{D}{2 a} z_{0} \cos θ}{1 - M_{x} \cos θ} - \frac{({mB}_{2} - {kB}_{1}) M_{x}}{z_{0}}] \frac{FA}{D}

(5)

The upstream rotor formula is obtained by swapping the indexes 1 and 2. The complete acoustic response of the CROR is then $p (x, r, θ) = p_{R_{1}} (\vec{x}, t) + p_{R_{2}} (\vec{x}, t)$ . In equations (4) and (5), MCA and FA are offsets representing the sweep and lean of the blade, respectively. Aside from the geometric definition of the configuration, the inputs of the model are the unsteady blade load distributions along the chord, expressed as an integral lift/drag coefficient $C_{L / Dk}$ multiplied by a normalized distribution $Ψ_{L / Dk}$ . This is actually the true source term in Hansonʼs model. Yet, in its original formulation, Hanson considered that the pressure variation along the chord was in phase, yielding real value for $Ψ_{L / Dk}$ , thus neglecting the time lag induced by the propagation of the flow structure along the blade. This was corrected here by expressing the distribution $Ψ_{L / Dk}$ as complex values, given by simple time and space Fourier decompositions of the unsteady pressure distribution over the blade. In this study, the harmonic pressure distribution over the blades are extracted from the unsteady numerical simulation. The extended Hansonʼs model has been compared with the analytical model recently proposed by Carazo et al.¹⁷

Similar Overall Acoustic Sound Pressure Level (OASPL) and reasonable agreement on the tonal directivities was found by Soulat et al. given the uncertainty on the blade loading distribution.⁷

Ffowcs-Williams and Hawkings’ analogy

In the second method, the far-field acoustics is obtained from the near-field unsteady conservative variables by a classical Ffowcs-Williams and Hawkingsʼ analogy (FWH). An extension of the code FoxWHawk (former Advantia) originally developed by Casalino¹⁸ is used for the present study.^19–22 Its formulation is based on the forward-time formulation for moving observers developed by di Francescantonio²³ on porous integration surface. It enables to perform on-the-fly acoustic calculations. On the contrary, a retarded-time approach would need the complete sampling of the solution to perform the acoustic extrapolation. For a fixed observer at $\vec{x} = (x_{1}, x_{2}, x_{3}) = (x_{i})$ , this yields the following expressions for the far field acoustic pressure:

4 π p (\vec{x}, t) = \frac{1}{c_{0}} \int_{S} [\frac{{\overset{\cdot}{F}}_{r}}{r (1 - M_{r})^{2}}]_{ret} dS + \int_{S} [\frac{F_{r} - F_{M}}{r^{2} (1 - M_{r})^{2}}]_{ret} d S + \frac{1}{c_{0}} \int_{S} [\frac{F_{r} (r {\overset{\cdot}{M}}_{r} + c_{0} (M_{r} - M^{2}))}{r^{2} (1 - M_{r})^{3}}]_{ret} d S

(6)

where

\vec{F}

is the pressure force acting on the surface S. F_M is this force projected in the source movement direction (equations (19) and (20) given by Casalino¹⁸). Only results from solid surface sources are presented below.

Results

Flow-field analysis

For the considered approach condition, there is a contraction of the streamlines downstream of each rotor (“vena contracta” at approach), but without any flow separation on the blades as shown by the mean velocity components in a meridional plane in Figure 3. The mean radial velocity is negative in between the two rotors and both axial and tangential velocities suggest a contraction of the front-rotor wake. The mean axial velocity also suggests a thick boundary-layer development on the long hub upstream of the front rotor. Moreover, given the torque balance between the two rotors and the clipping of the second rotor, the aerodynamic load is higher on the aft blades. The second rotor operates properly, and reduces the swirl generated by the front rotor. Yet, an important positive swirl downstream the tip region of the aft rotor is generated corresponding to the aft tip vortex.⁷

Figure 3.

Meanvelocity components in a meridional plane. (a) Blade to blade view (75% height)—vorticity and (b) flow structures—iso-Q factor.

The unsteady flow structures over most of the blade span are mainly characterized by the wake impingement of the front rotor on the aft rotor. A vorticity map in a blade to blade section clearly reveals the classical “chopping” of the front wakes by the aft blades (Figure 4). When convected downstream, the front wakes progressively merge with the aft wakes yielding the checkerboard pattern observed downstream. As pointed out by the grid study of Soulat et al.,⁷ the present fine simulations are accurate enough to properly propagate these flow patterns far downstream. It should also be emphasized that for these particular approach conditions, the wakes of both rotors are almost perpendicular and that any RANS simulation (frozen-rotor or mixing plane) would yield unphysical flow patterns in the aft rotor passage and downstream of the CROR, which may question any performance prediction on the aft rotor with a steady approach.

Figure 4.

Flow topology and vortical structures.

The flow topology is also dominated by several strong vortical structures. Besides the front-rotor wake, the most important ones are the two tip vortices generated by the rotors. In Figure 4(b), iso-surfaces of the Q-criterion reveal that the tip vortex coming from the front rotor impinges on the second rotor at approximately 87% of the span. This is to be contrasted with the results of Colin et al. at take-off conditions, which showed that the rear rotor cropping (similar to the present CROR) was sufficient to prevent the front-rotor tip vortex from interacting with the aft blade.⁶ An additional strong noise source is thus expected at approach. As for the viscous wake in Figure 4, the front-rotor tip vortex is thus chopped, but the resulting pieces keep some vortical structures as illustrated by the small patches of iso-Q surfaces visible downstream of the aft rotor. The aft tip vortex is even bigger and stronger, which is consistent with the higher load on this rotor. As shown in Figure 5, it initially has a wavy trajectory induced by the remaining of the front-rotor tip vortex travelling below it. It becomes a thick vortex sheet further downstream when all the front and aft tip vortex cores merge. The strength of this vortex sheet could be an additional strong noise source when interacting with the downstream empennage depending on the CROR installation on the plane.²⁴ Finally, a significant horseshoe vortex developing on the front rotor induced by the boundary layer developing on the long hub, outlined by an arrow in Figure 4(b), is mainly visible on the suction side of the blades. Although it decreases quite rapidly and completely disappears after impinging the aft blades, the vortex is still visible up to the leading edge of the second rotor and causes a third strong interaction with the aft rotor blade. Figure 5 also suggests that the front-rotor tip vortex does not have the same trajectory as the front-rotor wake, and that it will diffuse azimuthally. The grid study of Soulat et al.⁷ showed that the coarse mesh was responsible for an early dissipation of the vortical structures (the tip vortex disappearing 30% faster), and a transverse growth of these vortices with a 20% larger aft tip vortex. Nevertheless the mesh refinement does not alter the global characteristics of the flow significantly, such as the contraction behind the rotors and the height of the vortex impingement on the aft rotor.

Figure 5.

Contours of vorticity stressing the tip-vortex trajectory and evolution.

Noise source analysis

An accurate determination of the acoustic sources is of prime importance for a correct noise prediction. Interestingly, the harmonic load distributions e.g. the sources, have to be determined for noise prediction using equations (1)–(3), and are therefore directly available. The first four loading harmonics, thereafter noted $0 BPF, 1 BPF$ or BPF, $2 BPF$ , and $3 BPF$ are presented in Figure 6 for each rotor blade. When necessary (for the B₁–B₂ configuration), a subscript on these harmonics indicates if it refers to the front rotor (1) or to the rear rotor (2). The frequencies used for the harmonic decomposition are the blade passing frequency (BPF) and following sub-harmonics $({kB}_{1} Ω_{1} + {mB}_{2} Ω_{2}), \forall k, m \in ℤ$ . Load harmonic $0 BPF$ is the steady loading of the blade. Although constant, $0 BPF$ is responsible for the rotor loading noise at frequencies ${kB}_{1 / 2} Ω_{1 / 2}, \forall k \in ℤ$ because of the unsteadiness introduced by the blade movement. Soulat et al. showed that all higher harmonics had the same shape and order of magnitude as the third harmonics $3 BPF$ .⁷ Grid independence was also shown on the fine grid for these first harmonics. This is consistent with the harmonic decay found by Colin et al. at take-off conditions, where they showed similar low, almost constant levels beyond the third loading harmonics (Figure 13 in Colin et al.⁴)

Figure 6.

Dipolar source distribution over the blades. Harmonics ranked from 0 to 3.

As expected, the steady loading $0 BPF$ is mainly concentrated near the leading edge of each blade, and the aft rotor loading is higher than the front one. Moreover, the amplitude of the steady loading is a 100 times larger than the following loading harmonics.

Considering the first truly unsteady load harmonic ( $1 BPF$ ), the maximum pressure fluctuation over the aft rotor is about 10 times stronger than on the front rotor. This confirms that the wake and vortex impingement on the aft blade are far more energetic than the interaction of the potential effects with the front rotor. The fluctuations on the aft rotor are located mainly near the leading edge, except in the root and tip regions. This suggests that the wake intensity decreases quickly after the impingement, which is consistent with analytical models such as those of Sears^25,26 and Amiet,²⁷ and numerical results obtained at take-off by Colin et al. (Figure 11 in Colin et al.⁴). The effects of the front tip vortex are particularly clear here, maximizing the pressure fluctuation above 80% of the span. The tip vortex interaction is also responsible for a spread of the pressure fluctuations over the whole chord in the tip region. The impact of the small horse-shoe vortex is also visible, with another strong fluctuation-increase near the blade root. Although weaker (maximum pressure fluctuation 5 times weaker), the potential effects on the front blade are not negligible either. The unsteady loading topology is somehow different for this rotor, with a better spread of the fluctuations all over the blade surface. The maximum fluctuation is located in the upper half of the blade, at mid-chord. At the hub, the horse-shoe vortex developing from the front blade leading edge provides the fluctuations seen at the blade foot.

The fluctuations associated with the second loading harmonic $2 BPF$ are decreasing progressively compared with $0 BPF$ and $1 BPF$ . Again, the levels seen on the aft rotor remain much higher than on the front rotor, and the potential effects are still visible on the front blade. The topology on the front rotor is slightly different from the one observed for $1 BPF$ , with the maximum fluctuations located above 60% span, near the blade trailing edge. The global topology on the aft rotor is the same, but some small differences appear. The impact of the tip vortex now generates a maximum fluctuation near the leading edge only, quickly decreasing along the chord.

The rotor–rotor interactions can be traced up to the third order for the aft rotor and to the second order for the front one. Nevertheless, Soulat et al. noted that the impacts of the vortices on the aft blades are still clearly visible up to $5 BPF$ , confirming that these interaction mechanisms are probably the most effective from an acoustical point of view.⁷ These loading harmonics then provide the far-field acoustic pressure shown in the next section. Soulat et al. also showed that the mesh refinement had a strong impact on the unsteady fluctuations, with $1 BPF$ amplitudes up to 50% lower on the coarse mesh, particularly in the tip region. This emphasizes that the coarse mesh could be a good compromise for performance prediction of the machine (thrust, efficiency, etc.), but is definitely not acceptable for the noise predictions.

Radiated noise

Numerical assessment

To assess the quality and the robustness of each acoustic model, the impact of parameters such as mesh refinement, data sampling, and number of blade passing periods (BPP) required were studied for both models in details by Soulat et al.⁸ Only the main conclusions are recalled here. All directivities are considered as in the NASA experiment, which corresponds to a linear array of microphones as in the fly-by mode shown in Figure 7. The corresponding directivity angle θ is defined around a point on the axis half-way between the two rotors. It is zero along the rotational axis in the upstream direction and 180° along the rotational axis in the downstream direction.

Figure 7.

Acoustic experimental set-up.

For both methods, the mesh refinement had a strong effect both on the OASPL and the interaction tones. The modified Hansonʼs model was mostly found sensitive to the former whereas the FWH computations were effected by both. Going from the present 5 million node mesh to a finer one (about 8 million nodes), Soulat et al. noted that the OASPL were unchanged and that only small differences in the minima of the tone directivities (2–3 dB) were observed.⁸ This is consistent with the differences observed by Colin et al. at take-off when they varied both the mesh topology and the grid refinement (Figures 11 and 16, respectively, in Colin et al.⁵). The choice of the time discretization was also studied. A time-step of 75 samples per BPP was found necessary even though it triggered significant overhead in the FWH calculations. Any reduction brought significant speed-up but yielded noticeable modifications of the tone directivities, which is consistent with the recent study by Cunha and Redonnet.²⁸

The final parameter that was studied was the number of BPP. Significant differences were found between the two acoustic methods. On the one hand, a good convergence is achieved with only four BPP for Hansonʼs model for both the OASPL and the interaction tones. On the other hand, the FWH computations require much more BPP to resolve the phase information along the blade correctly. Convergence on the OASPL is reached for 12 BPP, whereas the low-frequency tones still have significant variations after 21 BPP. For instance, the low-frequency tone ${BPF}_{2}$ shown in Figure 8(a) has large spurious side-lobes for low BPP with some minima where Hansonʼs numerical result has a single maximum. Only a much longer time record (beyond 12 BPP) yields the expected correct large lobe centered around θ = 90°, and even with 21 BPP the side-lobes have not yet completed disappeared. Moreover, the spectra at a given position in Figure 8(b) clearly show that the longer the time record is (and consequently the smaller the frequency resolution is), the more tones are resolved, which is key for CROR tonal noise prediction. Even with 21 BPP, for which the frequency resolution is around 15 Hz, the two selected microphones show that new tonal peaks corresponding to interactions tones start being captured. Soulat et al. also looked at a B₁–B₁ configuration.^7,8 A much faster convergence of the FWH formulation was obtained as only 10 BPP were needed to capture the OASPL and the main tones, and 21 BPP yielded a proper resolution of the interaction tones.

Figure 8.

Convergence of the FWH noise prediction. (a) Directivity of mode BPF₂; (b) far-eld acoustic spectra at 90.

Validation

Soulat et al.⁷ first made comparisons with the results of Schnell et al.^7,29 The present B₁–B₂ configuration is indeed close to this other CROR considered by DLR in similar flight conditions, for which sideline acoustic directivities are also available. The far field noise was predicted using the modified Hansonʼs model (equation (1)), and compared with the results presented by Schnell et al. in Figure 9.²⁹ Compared with the experimental and numerical results obtained at DLR/TsaGi, the present method predicts the directivity of the two interaction tones ${BPF}_{1} + {BPF}_{2}$ and $2 {BPF}_{1} + 2 {BPF}_{2}$ fairly well. The only differences are an increased number of lobes for $θ \geq$ 100° and slightly more amplitudes of the dips. This might be caused by a slightly different blade loading.

Figure 9.

Comparison with the results of Schnell et al. (2012). (a) Current study; (b) From Schnell et al (2012).

The numerical results were then compared with the experimental data obtained on the present configuration, during the ORPR tests in the NASA LSWT wind tunnel. In Figure 10, the prediction of the modified Hansonʼs model are in good agreement with the measurements for all tones both in terms of levels and shape. For the ${BPF}_{1} + {BPF}_{2}$ tone, the only major difference seems to be around 30° where the model overestimates the level and has an extra lobe. Yet, the results of the FWH computations are very similar suggesting that it might come from the numerical prediction of the blade loading or some spurious installation effect in the experiment. For the second interaction mode ${BPF}_{1} + 2 {BPF}_{2}$ , both acoustic methods agree fairly well with the measurements. Yet, there is a slight off-set of the radiation lobe around 40–50° between the two simulations that the experimental data can hardly resolve. The amplitudes of the lobes centered at 110° and 140° are also reversed between the two predictions, Hansonʼs model providing a closer agreement with experiment. For the third interaction mode $2 {BPF}_{1} + {BPF}_{2}$ , the time resolution might not yet be enough to resolve all the lobes correctly in the FWH calculation at low and high θ angles, but the experimental trend is already recovered and its directivity can be clearly distinguished from the ${BPF}_{1} + 2 {BPF}_{2}$ tone. On the contrary, Hansonʼs prediction recovers the levels and its shape goes through the experimental scatter (sign of the difficulty to isolate this tone experimentally). For the highest-frequency interaction tone in Figure 10, $2 {BPF}_{1} + 2 {BPF}_{2}$ , both acoustic methods again agree fairly well with the measurements. Hansonʼs prediction yields the correct shape and levels suggesting that the source predictions are accurate and the slight shift at high θ angles in the FWH prediction might still be a time convergence issue. Colin et al. also recovers this interaction tone by a FWH calculation but with a larger simulation time (Figure 16 in Colin et al.⁵) but on a slightly different CROR configuration ( $B_{2} = B_{1} - 3$ ).

Figure 10.

Comparison of FWH–Hanson experiments on the same B₁–B₂ configuration. (a) Interaction mode BPF₁ + BPF₂; (b) Interaction mode BPF₁ + 2BPF₂; (c) Interaction mode 2BPF₁ + BPF₂; (d) Interaction mode 2BPF₁ + 2BPF₂.

Figure 11.

Comparison of far-field FWH acoustic spectra with NASA measurements.

FWH spectra at 21 BPP are also compared in Figure 11 with the measured spectra for three different microphone positions at 30°, 90°, and 139°, respectively. The harmonics $4 {BPF}_{1}$ and the interaction mode ${BPF}_{1} + {BPF}_{2}$ have the correct levels in all directions. Some higher-frequency interaction tones such as $2 {BPF}_{1} + 2 {BPF}_{2}, 4 {BPF}_{1} + {BPF}_{2}$ and ${BPF}_{1} + 5 {BPF}_{2}$ have also proper levels in most directions. The best predicted overall spectrum is at 90°.

Parametric study and further analysis

Using the modified Hansonʼs model, the effects of a variation of the blade numbers can be simply evaluated, as long as this variation does not affect the flow on the rotor significantly (e.g. a low solidity variation). As mentioned in “Numerical assessment” section, the noise emission was also calculated for B₂ = B₁ with a much faster time convergence for the tones. The OASPL directivity and the directivity of a few modes for both configurations are given in Figures 12 and 13, respectively. As expected from the source analysis, the noise emitted by the aft rotor is louder than the front rotor by 20 dB in average for both configurations. In the B₁–B₁ configuration, both rotors emit on the CROR axis, which is consistent with the equal number of blades on each row. As a consequence, this configuration is also louder than the B₁–B₂ configuration. The OASPL directivity of the latter is also modified for directivity angles between 20° and 160°. For instance, the B₁–B₂ configuration makes more noise above the CROR at 90°.

Figure 12.

OASPL directivities.

Figure 13.

Modal sound directivities. (a) B₁–B₁ conguration; (b) B₁–B₂ conguration.

The detailed modal directivity of each rotor is considered in Figure 13 for both configurations. The rotor-alone tones for the front and rear rotors are visible as the symmetric lobe located around θ = 90° for both configurations. Once again, the aft rotor noise is much higher than the front rotor noise, which is consistent with the source contributions shown in Figure 6. If both contributions had been compared at the same frequency ${BPF}_{2}$ , as Soulat et al.⁷ did, only marginal differences (a few decibels only) would have been found between the two rotors. Yet, for both CROR configurations at approach, the dominant noise mechanism is the interaction noise that is much higher than the rotor-alone tonal noise. Major contributors are the interaction modes ${BPF}_{1} + {BPF}_{2}$ and $2 {BPF}_{1} + 2 {BPF}_{2}$ . On both interaction tones, for both configurations, the 20 dB difference between the two rotors is clearly visible, confirming that the wake and vortex interaction is the most important mechanism for noise generation. The directivity patterns have several lobes for each rotor. The modal analysis also confirms that the interaction modes are responsible for the acoustic emission along the rotational axis of the CROR in the B₁–B₁ configuration. Similarly, the interaction mode ${BPF}_{1} + {BPF}_{2}$ causes the extra noise seen in the B₁–B₂ configuration at 90°. Finally, noise emission is globally stronger upstream of the CROR, which is thought to be related to the source concentration near the leading edges of the blades.

Conclusion

A hybrid method combining a three-dimensional unsteady RANS simulation and two acoustic analogies has been successfully applied to a realistic clipped CROR at approach conditions. The first analogy is an advanced-time formulation of the Ffowcs-Williams and Hawkingsʼ analogy. The second one is a frequency formulation of Ffowcs-Williams and Hawkingsʼ analogy, originally developed by Hanson and extended to non compact chord length. The unsteady simulation has been achieved for a B₁ = B₂ configuration on one sector but the noise predictions have been achieved for both B₂ = B₁ and $B_{2} = B_{1} - 2$ rear rotor blades.

The unsteady flow field shows several strong vortical structures, namely the wakes of both rotors, the horse-shoe vortex at the blade root of the front rotor and the strong tip vortices of both rotors. The unsteady blade loading on the cropped aft-rotor consequently shows three zones of strong interaction: the tip region is strongly affected by the impact of the front-rotor tip vortex caused by the vena contracta at approach condition; the hub region also shows the strong impact of the horse-shoe vortex formed on the front-rotor; finally the rest of the blade exhibits strong fluctuations caused by the impact of the front-rotor wake. All these pressure fluctuations are however concentrated at the leading edges of the rear blades. The flow analysis also shows some significant potential effect of the aft rotor on the front rotor. Finally, the strength of the tip vortex of the rear-rotor caused by the high loading of the cropped blade yield a strong vorticity layer that could cause a secondary noise source by impingement on some downstream empennage or some strong contrail vortices.

The implementation of the frequency acoustic model has been successfully compared with a model recently developed by Carazo et al.¹⁷ Consistent results have also been obtained with acoustic measurements on similar configurations performed at DLR.²⁹ The mesh is seen to have a major impact on the acoustic levels and directivity shape predicted by the method, and a mesh of at least 5 million points should be used at approach for a single blade sector, which properly resolves the boundary layers on both rotors and the rotor wake interaction. The acoustic signature also confirms the different noise sources identified in the unsteady flow analysis and shows a dominant contribution of the aft rotor acoustic radiation. Very good agreement is achieved for both acoustic models with NASA measurements on the present B₁–B₂ CROR configuration at approach conditions even though the FWH approach would still require more simulation time than the current 21 BPP to properly resolve some additional interaction tones. Yet, the spectral comparisons with experiment are very favorable especially at 90° from the CROR. Finally, it should be stressed that from a design point of view, the modified Hansonʼs model seems to be much more computationally efficient as it requires far less BPP (and therefore less URANS computational time) to time converge and yield the proper tonal directivities.

Footnotes

Acknowledgment

We wish to acknowledge the technical support from Ecole Centrale de Lyon on TurbʼFlow and Compute Canada and the RQCHP (Réseau Québécois de Calcul Haute Performance) for providing the necessary computational resources.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

We wish to acknowledge the financial support of Snecma-Safran Group.

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