**3. Acoustic analysis theoretical background**

The acoustic analysis consists of four constitutive parts:


After performing a Fast Fourier Transformation the sound pressure can be described as sum of acoustical modes propagating along the duct at a specific frequency. These modes are able to propagate in circumferential as well as in radial direction depending on the cuton frequency.

#### **3.1. Phase averaging and adaptive resampling**

information about the acoustic measurement section is given in Moser et al. [6] and in

The measurement system is made up by 11 multichannel pressure transducers PSI 9016 with a total amount of 176 channels and an accuracy of 0.05% full scale for pressure measurements. Four National Instruments Field Point FP‐TC‐120 eight‐channel thermocouple input modules and one FP‐RTD‐122 resistance thermometer input module is used. **Table 3** shows the measurement uncertainties (within a 95% confidence interval) of the five‐hole probe meas‐ urements. These values contain an error due to the multi‐parameter approximation, random error and the systematic error of the pressure transducers. The difference between the positive and the negative direction is a result of the multi‐parameter approximation. The measurement uncertainties of the static pressure and the total pressure at test rig inlet as well as at stage inlet are ±1 mbar. Uncertainties for total pressure measurements up‐ and downstream of the TEGVs are also in the range of ±1 mbar. The overall uncertainty of the total pressure loss coefficient ζ is estimated to be about ±0.0014. The random fluctuation of rotational speed is below 0.2% of the current operating speed. Measurement uncertainty of the temperature measurement is about ±0.5 K. The day‐to‐day variation of the operating parameters such as pressure ratio, corrected speed, rotational speed, total pressure and temperature at rig inlet has been below

After performing a Fast Fourier Transformation the sound pressure can be described as sum of acoustical modes propagating along the duct at a specific frequency. These modes are able to propagate in circumferential as well as in radial direction depending on the cuton frequency.

**Ma** +0.006 -0.003 [/] *α* +0.5 -0.08 [°] **pt** +3.3 -3.0 [mbar] **p** +5.3 -5.2 [mbar]

**Table 3.** Measurement uncertainties of the five-hole probe.

**•** phase averaging and adaptive resampling;

**•** azimuthal and radial mode analysis; and

**•** modal decomposition;

**•** computation of sound power.

**3. Acoustic analysis theoretical background**

The acoustic analysis consists of four constitutive parts:

Faustmann et al. [7].

0.5%.

*2.3.1. Measurement uncertainty*

10 Recent Progress in Some Aircraft Technologies

In order to determine the acoustic effects a phase locked averaging is done. For the two‐spool rig the phase of one of the two rotors is chosen. A shaft encoder from the monitoring system generates a pulse per revolution signal indicating start and end of one revolution. Triggering the flow is performed according to the triple decomposition procedure, which characterize a single source of periodic unsteadiness [8].

$$p(t) = \overline{p} + \langle p \rangle + p'(t) \tag{1}$$

The time dependent pressure *p*(*t*) is composed as sum of the averaged pressure, the purely periodic component *p* associated with a coherent periodic structure and the random fluctua‐ tion *p* '(*t*) that is mainly associated with turbulence.

Each revolution is divided into a fixed number of samples in order to correct small rotational speed variations of the rotor shaft [9]. The average of the samples at the same phase gives the phase averaged values. That procedure is well established and allows the identification of structures correlated to the rotor rotational speed. For a two‐spool rig all fluctuations of flow quantities induced by the other rotor are removed. Also, depending on which trigger signal is used for the analysis acoustic effects from the HP‐rotor or the LP‐rotor can be determined.

#### **3.2. Modal decomposition**

The acoustic field at any circumferential position can be written as superposition of several space and time dependent sound pressure waves (acoustic modes). The propagation of that pressure waves is described by the linearised wave equation <sup>1</sup> *c* 2 ∂<sup>2</sup> *p* <sup>∂</sup> *<sup>t</sup>* <sup>2</sup> −*∆p* =0. The solution of that equation is represented by the general expression (given in many publications, e.g. [10–14]):

$$p\left(\mathbf{x}, r, \varphi, t\right) = \sum\_{m = -\nu}^{\nu} \sum\_{n = 0}^{\nu} \left(A\_{mn}^{+} e^{ik\_{m}^{+}x} + A\_{m}^{-} e^{ik\_{m}^{-}x}\right) f\_{mn}\left(\sigma\_{mn} \frac{r}{R}\right) e^{im\varphi} e^{i\alpha t} \tag{2}$$

*Amn* <sup>+</sup> , *Amn* <sup>−</sup> are the complex modal amplitudes of order (m,n). *kmn* + and *kmn* <sup>−</sup> represent the axial wave numbers, *ω* is the angular frequency. The modal shape factor *f mn* depends on the eigenvalues of the Bessel function *σmn* and the geometry given by the hub‐to‐tip‐radius ratio *r <sup>R</sup>* . *f mn* represents the solution of the Bessel differential equation, describing the radial acoustic field considering hard wall boundary conditions [15, 16] and is defined as:

$$\int\_{V} f\_{mn} \left( \sigma\_{mn} \frac{r}{R} \right) = \frac{1}{\sqrt{F\_{mn}}} \left( J\_{mn} \left( \sigma\_{mn} \frac{r}{R} \right) + \mathcal{Q}\_{mn} Y\_{mn} \left( \sigma\_{mn} \frac{r}{R} \right) \right) \tag{3}$$

*Jmn* and *Ymn* are the Bessel functions of mth order of first and second kind. *Qmn* is also an eigenvalue and *Fmn* is a normalisation factor transforming the system from an orthogonal to an orthonormal eigensystem [15, 16].

The axial wave numbers depend on the local flow properties, such as axial Mach number *M aax* and the swirl number Ω. For the following investigations the swirl is approximated by a rigid body rotation of a steady flow, which leads to a modification of the wave number definition [17]: *k* ˜ <sup>=</sup>*<sup>k</sup>* <sup>−</sup>*<sup>m</sup>* <sup>Ω</sup> *<sup>c</sup>* . The axial wave numbers are then calculated as follows:

$$k\_{\rm \,m}^{\pm} = \frac{\tilde{k}}{1 - Ma\_{\rm \,ac}^2} \left[ -Ma\_{\rm ac} \pm \sqrt{1 - \left( 1 - Ma\_{\rm \,ac}^2 \right) \frac{\sigma\_{\rm \,m}^2}{\left( k \, R \right)^2}} \right] \tag{4}$$

Physical and geometrical conditions allow only the propagation of a certain number of specific mode combinations (m,n) along the duct. The axial wave number *kmn* <sup>±</sup> has to be real; otherwise the result of Eq. (2) will yield to an exponential sound pressure decay if *kmn* ± is a complex number. The frequency at which a mode (m,n) is first able to propagate is defined by the cuton frequency:

$$\left(\text{(kR)}\_{\text{mn}} = \sqrt{1 - \text{Ma}\_{\text{av}}^2 \sigma\_{\text{mn}}} + \frac{\text{m}\Omega \text{R}}{\text{c}}\tag{5}$$

The swirl factor mΩR <sup>c</sup> shifts the cuton frequency to higher or lower values, depending on the sign of Ω. Or in other words, for a specific frequency modes to be cut on are also shifted to higher or lower azimuthal mode orders m, hence the propagating mode distribution becomes asymmetric. The specific modes propagating through a duct downstream of a turbomachine stage result from the rotor‐stator interaction and are specified by a simple mathematical relation proposed by Tyler and Sofrin [5]:

$$\mathbf{m} = \mathbf{h}\mathbf{B} \pm \mathbf{k}\mathbf{V}; k = \cdots, -2, -1, 0, +1, +2, \cdots \tag{6}$$

h represents the harmonic index (e.g. 1 for the first blade passing frequency, 2 for the second, etc.), and B and V are the number of rotor blades and the number of stator vanes, respectively. According to Eq. (1) it is possible to determine the interactions of the rotor with a complete vane assembly by simply superimposing the effect of the single event. For a stator‐rotor‐stator assembly the modes can be predicted easily when extending Eq. (6).

$$\mathbf{m} = \mathbf{h}\mathbf{B} \pm \mathbf{k}\_1 \mathbf{V}\_1 \pm k\_2 V\_2; k\_{1,2} = \cdots; -2, -1, 0, +1, +2, \cdots \tag{7}$$

#### **3.3. Azimuthal and radial mode analysis**

1

%

*Ma*

*ax*

mode combinations (m,n) along the duct. The axial wave number *kmn*

1

±

s

an orthonormal eigensystem [15, 16].

12 Recent Progress in Some Aircraft Technologies

[17]: *k*

frequency:

The swirl factor mΩR

relation proposed by Tyler and Sofrin [5]:

˜ <sup>=</sup>*<sup>k</sup>* <sup>−</sup>*<sup>m</sup>* <sup>Ω</sup>

*mn*

*mn mn mn mn mn mn mn*

*r rr <sup>f</sup> J QY R RR <sup>F</sup>*

*<sup>c</sup>* . The axial wave numbers are then calculated as follows:

= - ± -- - æ ö

*mn ax ax*

the result of Eq. (2) will yield to an exponential sound pressure decay if *kmn*

*<sup>k</sup> <sup>k</sup> Ma Ma*

*Jmn* and *Ymn* are the Bessel functions of mth order of first and second kind. *Qmn* is also an eigenvalue and *Fmn* is a normalisation factor transforming the system from an orthogonal to

The axial wave numbers depend on the local flow properties, such as axial Mach number *M aax* and the swirl number Ω. For the following investigations the swirl is approximated by a rigid body rotation of a steady flow, which leads to a modification of the wave number definition

<sup>2</sup> <sup>2</sup> 1 1

Physical and geometrical conditions allow only the propagation of a certain number of specific

number. The frequency at which a mode (m,n) is first able to propagate is defined by the cuton

2 mn ax mn m R (kR) 1 Ma

s

sign of Ω. Or in other words, for a specific frequency modes to be cut on are also shifted to higher or lower azimuthal mode orders m, hence the propagating mode distribution becomes asymmetric. The specific modes propagating through a duct downstream of a turbomachine stage result from the rotor‐stator interaction and are specified by a simple mathematical

h represents the harmonic index (e.g. 1 for the first blade passing frequency, 2 for the second, etc.), and B and V are the number of rotor blades and the number of stator vanes, respectively. According to Eq. (1) it is possible to determine the interactions of the rotor with a complete

é ù ê ú

ç ÷ ë û è ø

ss

( ) <sup>2</sup> 2

c

<sup>c</sup> shifts the cuton frequency to higher or lower values, depending on the

m hB kV; , 2, = ± = -- ++ *k* L L 1,0, 1, 2, (6)

W

=- + (5)

*mn*

% (4)

<sup>±</sup> has to be real; otherwise

± is a complex

*k R*

s

æö æö æö æ ö ç÷ ç÷ ç÷ = + ç ÷ èø èø èø è ø (3)

The computation of the propagating sound field in a turbomachine is based on the determi‐ nation of the amplitudes *Amn* ± in Eq. (2). That means that the sound pressure at several axial and circumferential positions has to be measured first, e.g. by means of a microphone array. Then the radial mode analysis (RMA) described, e.g. in Holste and Neise [13], Tapken and Enghardt [15], and Enghardt et al. [12, 18] is applied. The RMA is a methodology used for the modal decomposition of an in‐duct acoustic field. The data is used to reconstruct at specific times the instantaneous circumferential pressure distribution. The application of a spatial Fourier Transformation over this data set represents the first action and is called azimuthal mode analysis (AMA):

$$\mathbf{A\_{m}(x,r)} = \frac{1}{\mathbf{N\_{\varphi}}} \boldsymbol{\Sigma}\_{\mathbf{k=1}}^{\mathbf{N\_{\varphi}}} \mathbf{p(x,r,q\_{k})} \mathbf{e^{-im\varphi\_{k}}} \tag{8}$$

In order to determine the complex amplitudes *Amn* <sup>±</sup> at a specific angular frequency *ω* for each azimuthal mode m Eq. (2) can be written as [18, 19]:

$$\mathbf{A}\_{\rm m}\left(\mathbf{x},\mathbf{r}\right) = \sum\_{n=0}^{\prime\prime} \left( A\_{\rm m}^{+} e^{\beta \hat{k}\_{\rm m}^{+} \mathbf{x}} + A\_{\rm m}^{-} e^{\beta \hat{k}\_{\rm m}^{-} \mathbf{x}} \right) f\_{\rm m}\left(\sigma\_{\rm m} \frac{r}{R}\right) \tag{9}$$

For each azimuthal mode order m a linear equation system **Am** =**WmAmn** can be found. Since this equation system is (usually) highly overdetermined, a least‐mean‐square fit algorithm is used to solve that inverse problem.

#### **3.4. Computation of sound power**

Considering the energy carried by each individual mode in a hard walled duct, the effective sound power can be determined according to Morfey [17]:

$$A\_{\mathbf{m}}(\mathbf{x}, \mathbf{r}) = \Sigma\_{n=0}^{\circ \circ} \left( A\_{mn}^{+} e^{ik\_{mn}^{+} \chi} + A\_{mn}^{-} e^{ik\_{mn}^{-} \chi} \right) f\_{mn} \left( \sigma\_{mn} \frac{r}{R} \right) \tag{10}$$

The complex factor αmn = 1−(1−Maax <sup>2</sup> ) <sup>σ</sup>mn 2 (k˜R)2 contains the definition of the cuton frequency.
