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Numerical simulation of multistage turbine sound generation and propagation ✩ Numerische Simulation der Schallentstehung und – ausbreitung in einer mehrstufigen Turbine Detlef Korte ∗ , Thomas Hüttl, Fritz Kennepohl, Klaus Heinig MTU Aero Engines GmbH & Co. KG, Dachauer Str. 665, 80995 München, Germany Received 21 January 2004; received in revised form 13 December 2004; accepted 14 December 2004 Available online 19 January 2005

Abstract A numerical method has been developed to calculate the noise within a turbine or a compressor by means of CFD (Time Linearized Euler Code). The noise generation, caused by the interaction of neighbouring cascades, can be calculated as well as the noise propagation within the turbomachine. Numerical results obtained by this method are compared to measurements. The CFD results are qualitatively in good agreement with the measurements. For low azimuthal mode orders at BPF1 , the computed modal sound power levels at the turbine exit are about 5 to 10 dB too high compared with measurements further downstream in the exhaust duct. Especially the cut-on, cut-off behaviour of the modes in the duct is simulated successfully. 2005 Elsevier SAS. All rights reserved. Zusammenfassung Eine numerische Methode zur Berechnung des Lärms einer Turbine / eines Verdichters mittels CFD (Zeitlinearisiertes Euler Verfahren) wurde entwickelt. Sowohl die Lärmentstehung, verursacht durch die Interaktion benachbarter Schaufelgitter, als auch die Lärmausbreitung innerhalb der Turbomaschine kann berechnet werden. Numerische Ergebnisse, die durch diese Methode ermittelt wurden, wurden mit Messungen verglichen. Die CFD Ergebnisse sind qualitativ in guter Übereinstimmung mit den Messungen. Für niedrige Umfangsmodenordnungen bei BPF1 , sind die berechneten modalen Schallleistungspegel am Turbinenaustritt etwa 5 bis 10 dB zu hoch, verglichen mit Messungen an einer Stelle weiter stromab im Strömungskanal. Speziell das cut-on / cut-off Verhalten der Moden im Kanal wurde erfolgreich simuliert. 2005 Elsevier SAS. All rights reserved. Keywords: Time linearized Euler method; Multi-stage turbomachine; Sound propagation; Wake interaction; Noise reduction Schlüsselwörter: Zeitlinearisiertes Euler Verfahren; Mehrstufige Turbomaschine; Schallausbreitung; Nachlaufinteraktion; Lärmreduktion

1. Introduction

✩

This article was presented at the German Aerospace Congress 2003.

* Corresponding author. Tel.: +49 (0)89 1489 2111, fax: +49 (0)89 1489

96088. E-mail address: [email protected] (D. Korte). 1270-9638/$ – see front matter 2005 Elsevier SAS. All rights reserved. doi:10.1016/j.ast.2004.12.002

For a proper prediction of the sound emission of multistage turbomachines, both the generation and the propagation of sound through the cascades are important aspects. The generation of sound in turbines and compressors is driven by unsteady aerodynamic phenomena, mainly rotor-

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stator interaction, [28]. Several numerical methods have been developed in order to study wake or potential flow field interaction of adjacent cascades. Semi-analytic methods, like the lifting surface theory are fast tools for 2D or 3D investigations, if axial main flow is uniform and the stator vanes are thin and aligned with the main flow, [28– 31]. Time linearized Euler codes are very popular methods as they allow to investigate radially varying and swirling flow [5,6,9,26]. Early investigations of rotor-stator interactions using Navier–Stokes solvers have been restricted to two-dimensional methods [3,4,32,36]. Recently, research efforts have been made using 3D Navier–Stokes codes, but the computational requirements are very high, [8,35]. First attempts have been performed, to apply Large-Eddy simulation to rod-airfoil interaction [17] which is a preceding test case for rotor-stator interaction. The propagation of sound through cascades have first been studied by analytical methods [1,2,22,24]. Depending on the angle of incidence of incoming waves, cascades can have a shielding effect that reduces the sound emission significantly [30]. Apart from these single cascade effects, additional two- and multi-cascade effects occur, e.g. mode-trapping [11] or sum and difference tones due to scattering. Several analytical or numerical methods for calculating sound propagation through multiple cascades have been developed [2,10,12,13]. Some of these mostly analytical methods are restricted to two-dimensional flows through flat plates. The progress in computer technology now allows us also to use numerical finite volume/finite element methods to consider full 3D geometry and flow. Within the European research project TurboNoiseCFD, a methodology has been developed by MTU to calculate the tonal sound generation due to aeroacoustic interaction as well as the propagation of tonal sound through multistage turbomachines. Three rotorstator interaction mechanisms are considered: wake interference and upstream and downstream interference of the steady pressure field fixed to a cascade, interacting with a neighbouring blade row. The numerical method is based on the time linearized Euler code Lin3D for 3D cascade flows, developed at MTU. The unsteady flow is assumed to be a small harmonic perturbation of the non-linear steady flow, so that the steady flow problem is separated from the unsteady problem. As long as the wave amplitudes remain moderate, the higher order terms in the governing equations derived under this assumption can be neglected and the describing unsteady flow equations become linear. The small-disturbance Euler equations are transformed into the frequency domain, so that the results are given as amplitudes and phases of the flow variables in the whole (3D) computational domain. Lin3D is already used for aeroelastic design of aeroengine compressors and turbines [18–21]. The code and its extension to multistage sound generation calculations have also been tested for various aeroacoustic applications, ranging from

simplified flat plate test cases to realistic single- or multistage turbomachines [6,7,15,16,23]. Uslu et al. [33,34] performed a study on sound propagation in a 3-stage high speed low pressure compressor. Here, Lin3D was used for noise generation computations and TRLOSS for the sound propagation, based on the four pole theory coupling 2D flat plate calculations [13,14]. This method should preferably be used for compressors where the 3D geometry of blades and vanes is very close to 2D flat plates. Uslu et al. [33,34] successfully studied the effect of blade or vane number variations on sound generation and sound propagation. This method is not suitable to study the effect of blade/vane geometry variations (e.g. lean, sweep, bow) on sound propagation, because the midspan geometry, taken for the 2D flat plate calculations, does not change. The focus of the present study is a new method to calculate noise propagation within a turbomachine. This propagation method shall be applicable to compressors as well as turbines and may thus not be restricted to blade/vane geometries close to flat plates [25]. Furthermore, it is intended in the future to perform parametric investigations in order to study the effect of lean, sweep or bow on sound generation and sound propagation. The method of Uslu et al. [33,34] did not fulfil these challenging requirements any more.

2. Test case The test case of the present investigation is a threestage low pressure turbine, for which measurements have been performed by DLR within the earlier EC project “RESOUND”, see Fig. 1. The blade numbers are: 123 (Rotor 1), 77 (Rotor 2) and 74 (Rotor 3); the vane numbers are 38 (Stator 1), 117 (Stator 2) and 126 (Stator 3). The three rotors are mounted on the same shaft. Measurements have been performed within the RESOUND programme for configurations with and without exit guide vane (EGV). The data used for a comparison with

Fig. 1. RESOUND turbine, a 3-stage LP turbine.

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Fig. 2. Mean frequency spectrum of the RESOUND turbine without EGV (measurements by DLR).

the numerical results in this report were measured without EGV (Fig. 2). The pressure sensors for the noise measurements are placed downstream of the turbine (see Fig. 1). Numerically, the noise propagation was only calculated up to the exit of Rotor 3. The different position with respect to the measurement plane has to be taken into account when comparing numerical data and measurement data, especially because the radius decays from calculation plane (exit of Rotor 3) to measurement plane. The calculations were performed for the configuration without exit guide vanes at 74% design speed (cruise) in order to simulate approach conditions. This reduced shaft speed has been chosen, because it is relevant for the noise certification of the engine.

3. Numerical results 3.1. Generated sound The sound generation mechanism assumed to be dominant, is the interaction between neighbouring cascades. Due to the interactions, tones are generated at the blade passing frequencies (BPF) and their harmonics. At each cascade (stator or rotor) tones can be generated due to one of the following disturbances, caused by a neighbouring cascade, leading to an unsteady loading of the vanes or blades: (a) steady pressure field (“potential field”) of upstream cascade, rotating relatively to the considered cascade; (b) steady pressure field (“potential field”) of downstream cascade, rotating relatively to the considered cascade; (c) viscous wake of upstream cascade, rotating relatively to the considered cascade. Only these mechanisms are taken into account in the presented calculations. Other noise generation mechanisms

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Fig. 3. Sound power spectrum BPF1 (123) at location of noise generation, upstream travelling sound waves.

and combination tones have not been studied. Each interaction (a, b and c) is treated separately during the simulation process. Noise is generated and radiated into upstream and downstream flow directions. Accordingly the upstream and downstream sound is calculated for each interacting pair of cascades. Only tones below 10 kHz were calculated: BPF1 (123), BPF2 (77), BPF3 (74), 2BPF2 (77), 2BPF3 (74). Tones at frequencies of more than 10 kHz are practically irrelevant for noise certification due to high atmospheric damping at high frequencies. The calculations are performed on the finest of three computational meshes. Each calculation results in a sound power level distribution of azimuthal modes. The sound power is calculated by integrating the radially varying intensity over a duct crosssection. The intensity is given by the Morfey equation [27]. Because the different interaction mechanisms of neighbouring cascades are treated in separate calculations, there can be up to 3 different values of sound power levels for one azimuthal mode. The modal sound power spectra at frequency BPF1 are presented here as examples of the obtained results for acoustic waves travelling in upstream (Fig. 3) and downstream directions (Fig. 4). It turns out that for each frequency the sound power of up to five azimuthal modes is clearly higher than the sound power of the others. These are the main modes with respect to noise investigations. For BPF1 , the following modes are relevant: m1 = B1 − j V1 , with j = 2 (m1 = 47), j = 3 (m1 = 9), j = 4 (m1 = −29), j = 5 (m1 = −67), and m2 = B1 − j V2 , with j = 1 (m2 = 6). 3.2. Propagated sound It is not only important to know the sound power at its origin, but also at the exit of the turbine. Since no exit guide vane was present in the case investigated here, it was decided

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Fig. 4. Sound power spectrum BPF1 (123) at location of noise generation, downstream travelling sound waves.

to calculate the propagation of sound only up to the exit of Rotor 3 instead of calculating it up to the exit of the turbine (Fig. 1) to save computational time and because of numerical dissipation. The chosen method (“beetle method”)1 is a rather simple perturbation propagation: For each relevant azimuthal mode, the downstream (upstream) travelling pressure perturbation at the outlet (inlet) of one calculation is used as the inlet (outlet) perturbation of the downstream (upstream) following cascade, after a transformation to the relatively rotating coordinate system. The time-linearized Euler code Lin3D has been used for calculating the propagation of the perturbation through each cascade between sound source and turbine exit. A decomposition into radial modes is unnecessary for the calculation, but for postprocessing purposes a decomposition into radial modes using Bessel functions gives an additional insight into flow phenomena. The sound propagation of the three blade passing frequencies was calculated. Only straightforward sound propagation calculations in flow direction have been performed. No reflections or upstream propagation were considered. Azimuthal and radial mode scattering occurs when a perturbation mode is travelling through a cascade (see Section 4.3). Nevertheless, scattered modes were ignored in the present results of noise propagation. The calculations have been performed on the medium size mesh out of three meshes, because parametric studies have shown that the differences between the medium and the costly fine mesh results were not significant. As an example, the propagation of the relevant azimuthal modes of the first blade passing frequency (BPF1 ) is presented in Fig. 5. BPF1 has been selected as it allows us to look at both the generation process and the transmission through successive rotor and stator rows. The legend of 1 We called the propagation calculation method ”beetle method” in order to illustrate the stepwise calculation procedure from source to turbine exit.

Fig. 5. Sound propagation – CFD Results for BPF1 .

Fig. 5 contains the following information: mode order; location of sound generation; mechanism of sound generation. The generated modal sound power levels were presented in Fig. 4. For these modes (m = −67, −29, +6, +9, +47) sound propagation calculations have been performed up to the exit of Rotor 3. The resulting modal sound power levels at the exit of each cascade are presented in Fig. 5. The most left point of each curve denotes the sound power at the generation origin. Mode 47, generated at V1 by interaction with potential field of B1, is not shown because it is cut-off at the source (V1 exit). Two different effects can be observed. On the one hand some modes remain cut-on at all cascades (m = −29, +6, +9); on the other hand there are modes that switch between cut-on and cut-off (m = −67, +47). The sound powers of the cut-on modes (m = −29, +6, +9) decay monotonously from the sound generation origin to the last cascade due to transmission losses. The sound powers of the other modes (m = −67, +47) are represented by zigzag lines in Fig. 5. One cascade downstream of their generation origin they become cut-off, in the next stage cut-on and then cut-off again up to the exit of Rotor 3. The pressure disturbance amplitudes (not shown) of all modes decay monotonously from cascade to cascade due to transmission losses. However, pressure disturbance and sound particle velocity are almost in phase for a cut-on mode

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but if the phase difference is approximately 90◦ , the respective mode is cut-off. The increase in sound power from one cascade to the next, related to a change from cut-off to cut-on, is an effect that can be looked on as a sound generation process: The disturbance pressure field of an upstream cut-off mode can interact with the steady pressure field of the downstream cascade and thus noise can be generated, just as in the case of the potential field interaction described in Section 3.1. Please note that the range of cut-on modes is shifted due to swirl. In the present case negative mode numbers (e.g. −67) can become cut-on in the swirling flow between V2 and B2, whereas lower absolute, but positive mode orders (e.g. 47) are cut-off.

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of the same order, but generated due to the different mechanisms, are in phase. Thus, a “worst case” value of the radiated noise is calculated for each azimuthal mode. Only a few azimuthal modes of noise generation calculations and measurements correspond to each other. These corresponding modes are marked in Fig. 6. While measured values exist for a limited, but predominantly continuous, range of azimuthal mode orders, calculated values exist only for some discrete mode orders, generated by the interaction of neighbouring cascades. Sound power levels below 60 dB are not considered important and consequently not presented below. Values up to this limit can be the result of numerical effects during calculations. Comparing measured and calculated sound power levels of one azimuthal mode, three cases can be observed:

4. Comparison CFD vs. measurement 4.1. CFD results at generation origin vs. measurement at downstream position A comparison between the measured and the calculated sound power values at source is presented in this section to show that propagation effects cannot be neglected. The calculated modal sound power levels are compared with the measured values In Fig. 6. The diagram on the left hand side shows the calculated sound power levels of the frequency BPF1 at the sound generation location. The diagram on the right hand side shows the measured sound power levels of the same frequency BPF1 downstream of the turbine. The modal sound power levels of Fig. 6(a) are the result of a summary of the sound power levels presented in Fig. 4, in the sense that the sound powers of the same azimuthal mode order, but generated due to different mechanisms, are summed up. For this summation it was assumed that modes

(1) the modal sound power level exists in both cases and its value exceeds the lower limit specified above (>60 dB); (2) the modal sound power level does not exist in case of the calculation; the modal sound power level exists in case of the measurement and exceeds the lower limit, specified above (>60 dB); (3) the modal sound power level exists in case of the calculation and exceeds the lower limit, specified above (>60 dB); the modal sound power level does not exist in case of the measurement or is below the limit (<60 dB). Considering these three cases the following can be stated concerning the BPF1 results: (ad 1) The three azimuthal mode orders m = −29, +6, +9 exist as well in the noise generation calculations as in the downstream measurement results (Fig. 6). They are marked by green colour in Fig. 6(a) and arrows in Fig. 6(b).

Fig. 6. (a) CFD results for sound power spectrum (BPF1 ) at generation origin. (b) Measured sound power spectrum (BPF1 ) downstream for the corresponding test case “C0, OP2” at DLR.

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Accordingly it is assumed that the calculated modes m = −29, +6, +9 will be propagated (cut-on) up to the measurement plane. This is shown to be true (at least up to the exit of Rotor 3) in Fig. 5. In the measurement, the sound power levels of the three common modes are high, relative to values of similar mode orders. Thus, one can clearly see peaks in Fig. 6(b) at these mode orders. This indicates the importance of the noise generation mechanism exclusively investigated in the calculations, i.e. the interaction of neighbouring cascades. Nevertheless, the difference of the absolute values, e.g. 17 dB in case of mode order m = +9, is very high when the measurement is compared to the calculated results at the source. Beyond that, the difference in case of mode m = +9 varies strongly from the differences in case of modes m = −29 and m = +6. The last two topics indicate the importance of the propagation effects. These cannot be neglected. (ad 2) The existence of peaks, like the one for mode m = −32 in the measurement spectrum, Fig. 6(b) (no correspondence in the calculation spectrum, Fig. 6(a)), shows that other noise generation mechanisms can be important too, as for example the interaction of three cascades. These are not modelled numerically. (ad 3) A peak in the generation spectrum, Fig. 6(a), without a corresponding one in the measurement spectrum, Fig. 6(b), leads to the assumption that this mode will change to cut-off during the propagation calculation from the noise generation origin to the measurement plane. The highest peak of such a mode corresponds to the order m = 47 (see Fig. 6(a)). The measured modal value is very low, such that it is not even visible in Fig. 6(b). It is shown in Fig. 5 that mode 47 indeed becomes cut-off during the propagation calculation. Investigations of the other blade passing frequencies show similar behaviour. 4.2. CFD results of sound propagation to Rotor 3 exit vs. measurement at downstream position In this part the sound power levels of the BPF1 azimuthal modes, propagated to the Rotor 3 exit, are compared to experimental values, measured further downstream. The relevant modes from the noise generation calculations have been used to start noise propagation calculations up to the exit of Rotor 3. This is not the measurement plane. Propagation calculations up to the measurement plane have not been conducted. No reflection was considered, only straightforward propagation of the disturbance modes was conducted. Five azimuthal modes of the frequency BPF1 have been propagated straightforward. During the process, parts of the sound power of a specific azimuthal mode and frequency are transferred to other modes and frequencies. The evaluation within this section only considers the sound power related to the original disturbance mode and frequency.

The propagation calculations of these modes (m = −67, −29, 6, 9, 47) were presented in Section 3.2. In this part they are compared to measured values. The results of the comparisons of the modes generated due to different mechanisms are presented in Fig. 7. The sound powers of the same azimuthal mode order, but generated due to different mechanisms, are summed up to total sound power levels and compared to the measured values in Fig. 8. A blue bar denotes the calculated-generated noise, a red bar the calculated-propagated noise and a yellow bar the measured sound. The sound of mode −67 is generated at Stator 1, due to the pressure field of rotor 1 (see Fig. 4). The sound propagation calculations were presented in Fig. 5. The calculated sound power level at the exit of Rotor 3 is compared to the measured value in Fig. 7(a) and Fig. 8. The sound of mode −29 is generated by three mechanisms: at Stator 1, due to the pressure field of Rotor 1; at Rotor 1 due to wake and pressure field of Stator 1 (see Fig. 4). The sound propagation calculations of all three disturbances were presented in Fig. 5. The calculated sound power levels at the exit of Rotor 3 are compared to the measured value in Fig. 7(b). Three mechanisms of noise generation lead to a sound mode of order m = 6. Two of them are generated on Stator 2, one due to the wake of Rotor 1, the other due to the pressure field of Rotor 1. The third is generated at Rotor 1, due to pressure field of Stator 2 (see Fig. 4). All of them were selected as disturbances for propagation calculations (see Fig. 5). The comparison of the propagation results to the measured value is presented in Fig. 7(c). Three mechanisms of noise lead to a sound mode of order m = 9 (see Fig. 4): noise generation at Stator 1 due to the pressure field of Rotor 1; noise generation on Rotor 1 due to wake and pressure field of Stator 1 (see Fig. 4). Although one of them is clearly dominant, all were selected as disturbances for propagation calculations (see Fig. 5). The comparison to the measured value is presented in Fig. 7(d). Two mechanisms of noise generation lead to a sound mode of order m = 47 (see Fig. 4). Both of them were selected as disturbances for propagation calculations (see Fig. 5): both tones are generated at Rotor 1, one due to the pressure field of Stator 1, the other due to the wake of Stator 1. The comparison to the measured value is presented in Fig. 7(e). Except for mode −67, the calculated sound power level at the exit of Rotor 3 is always slightly greater than the measured one. The calculated value for mode −67 is much lower than the measured one. Nevertheless they are at least consistent since they are low with respect to measured sound powers of other modes (see Fig. 8). The measured sound power level of mode 6 is the highest of all modes while the measured value of mode 47 is the lowest of all considered five modes. In contrast, both noise generation calculations lead to noise levels in the same range. It is shown in Fig. 7(c, e) and Fig. 8 that the sound

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Fig. 7. Sound power levels for BPF1 – modes: −67, −29, 6, 9, 47 – calculated propagation vs. measurement.

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Fig. 8. BPF1 – Spectrum of modal sound powers – Calculation vs. measurement.

power levels of these modes, after propagating them to the exit of Rotor 3, match well with the measured values. It was shown in Fig. 5 that the sound power of mode 6 decays during the propagation, but the mode remains cut-on. It was also shown that mode 47 becomes cut-off at the exit of Rotor 3. Accordingly its sound power decays nearly to zero. This corresponds to the low measured value. The behaviour of the modes m = −29, +9 was shown to be analogous to mode 6 (see Fig. 5). The sound power decays monotonic from one stage to the next. At the exit of Rotor 3, the sound power level matches well with the measured value in case of mode 9 (see Fig. 7(d) and Fig. 8). The behaviour of mode m = −67 was shown to be analogous to mode 47 (see Fig. 5). It becomes cut-off at the exit of Rotor 3, which is related to a sound power level near zero. Fig. 8 shows, that the CFD results are qualitatively in good agreement with the measurements at BPF1 . For low azimuthal mode orders, the computed modal sound power levels at the turbine exit are about 5–10 dB too high compared with measurements further downstream in the exhaust duct. Please note that the modal sound power levels are compared at two different positions in the duct. While propagating, the noise might have been reduced in the experiment by physical damping due to nonlinear effects and turbulence. 4.3. Mode scattering effects in azimuthal and radial order Another interesting effect observed when doing a propagation calculation with the presented method is the mode scattering in azimuthal and radial order. Azimuthal mode scattering occurs when a mode m interacts with a cascade (e.g. rotor with blade number B). Additional modes of mode order m = m ± j B (with: j integer) can occur. This is illustrated in the example of azimuthal mode m = 6, generated at Stator 2 and prescribed as a disturbance at the following

cascade, Rotor 2 (Fig. 9). Apart from the sound pressure at mode 6, high sound pressures occur at the azimuthal mode m = +83, a lot less at other mode orders (azimuthal mode scattering). Beyond that, the radial decomposition of mode 6 using Bessel functions leads to different results at the entrance (disturbance) and at the exit of the grid (radial mode scattering). The effects of mode scattering can be very different for different azimuthal modes. While for one mode almost no sound power is shifted, almost the total sound power can be transferred in other cases. This is shown in Fig. 10. Mode 6 and mode 47, both generated on Rotor 1, are prescribed as pressure disturbances at the entrance of Stator 2. The scattering effects are minimal when propagating mode 6. In case of mode 47 a different behaviour can be observed: most of the sound power is transferred to mode −70, which is cut-on at the exit of Stator 2, while mode 47 becomes cut-off.

5. Conclusion The noise of a turbine has been calculated by means of CFD. This paper presents the results of noise generation and propagation calculations using a linearized Euler code (Lin3D). These results are compared with measured data, delivered by DLR. Consistencies and differences between simulation and measurement are described. The CFD calculations consider the noise generated by the interaction of two neighbouring cascades. The comparison of numerical results and measured data shows that this noise generation mechanism is indeed very important. But it also shows that other mechanisms should not be neglected. One of such mechanisms is the interaction of three cascades. It has been shown in this paper that it is important to perform the noise propagation calculations up to the last stage,

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Fig. 9. Circumferential and radial scattering at Rotor 2 of mode m = +6 (generated at Stator 2); disturbance mode at entry of Rotor 2, all other values at exit of Rotor 2.

Fig. 10. Circumferential mode scattering.

even if a mode becomes cut-off at an intermediate stage. Indeed it can become cut-on again. To propagate the sound of a mode from one cascade to the next, its pressure disturbance evaluated in the upstream cascade exit was prescribed at the inlet of the investigated grid (‘Beetle Method’). Such a disturbance pressure field does also exist in case of a cut-off mode. Its amplitude would decay to zero over some distance but in case of the RESOUND turbine the stages are very close together. The propagation of a cut-off mode can also be looked at as a noise generation process: The steady pressure field of the investigated cascade interacts with the relatively rotating disturbance pressure field and such generates tones, just as in the potential field interaction process. The calculated sound power of a mode, which is cut-on at the exit of the last stage, seems to be reasonable, even if the mode was cut-off at an intermediate stage. While another tested method of sound propagation [33, 34], the four pole theory [13,14], is exact under certain cir-

cumstances (e.g. no mode conversion), the beetle method leads to better results when any reflection of waves can be neglected. In practice, reflection of waves could become very relevant for multistage turbomachines. Phenomena of an infinite number of reflections of sound waves between two cascades can be important. In that case, the propagation of a single acoustic mode depends on the axial gap between two cascades and – compared with computations neglecting reflections at all – the sound can be amplified or extinguished when superposing the sound waves. A first improvement of the beetle method could be to consider in turbine or compressor cases the first reflection of the upstream (downstream) propagating sound at the neighbouring cascade. Then, direct downstream (upstream) propagating sound and reflected downstream (upstream) propagation sound have to be superposed. Comparing the computed and measured absolute sound power levels, the following uncertainties have to be considered: (i) The position of the plane, up to which the noise

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propagation is calculated, is different from the measurement plane. (ii) In the simulation, one mode can be created by several disturbances (wake, potential pressure field interaction). To calculate the resulting sound power level, the phase relation would have to be considered. This was done in this investigation by a worst case consideration (i.e., summation in phase for each mode).

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Acknowledgements The support of the DLR, who provided the evaluated measurement results, is gratefully acknowledged. The work described in this paper has been carried out in the framework of the research project ‘Turbomachinery noise source CFD models for low noise aircraft designs – TurboNoiseCFD’ funded by the European Commission within the GROWTH – 5th framework programme.

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