# Acoustics/Rotor Stator Interactions

An important issue for the aeronautical industry is the reduction of aircraft noise. The characteristics of the turbomachinery noise are to be studied. The rotor/stator interaction is a predominant part of the noise emission. We will present an introduction to these interaction theory, whose applications are numerous. For example, the conception of air-conditioning ventilators requires a full understanding of this interaction.

## Noise emission of a Rotor-Stator mechanism

A Rotor wake induces on the downstream Stator blades a fluctuating vane loading, which is directly linked to the noise emission.
We consider a B blades Rotor (at a rotation speed of $\Omega$) and a V blades stator, in a unique Rotor/Stator configuration. The source frequencies are multiples of $B\Omega$, that is to say $mB\Omega$. For the moment we don’t have access to the source levels $F_{m}$. The noise frequencies are also $mB\Omega$, not depending on the number of blades of the stator. Nevertheless, this number V has a predominant role in the noise levels ($P_{m}$) and directivity, as it will be discussed later.
Example
For an airplane air-conditioning ventilator, reasonable data are :
$B=13$ and $\Omega =12000$ rnd/min
The blade passing frequency is 2600 Hz, so we only have to include the first two multiples (2600 Hz and 5200 Hz), because of the human ear high-sensibility limit. We have to study the frequencies m=1 and m=2.

## Optimization of the number of blades

As the source levels can't be easily modified, we have to focus on the interaction between those levels and the noise levels.
The transfer function ${{F_{m}} \over {P_{m}}}$ contains the following part :
$\sum \limits _{s=-\infty }^{s=+\infty }{e^{-{{i(mB-sV)\pi } \over 2}}J_{mB-sV}}(mBM)$
Where m is the Mach number and $J_{mB-sV}$ the Bessel function of mB-sV order. In order to minimize the influence of the transfer function, the goal is to reduce the value of this Bessel function. To do so, the argument must be smaller than the order of the Bessel function.
Back to the example :
For m=1, with a Mach number M=0.3, the argument of the Bessel function is about 4. We have to avoid having mB-sV inferior than 4. If V=10, we have 13-1x10=3, so there will be a noisy mode. If V=19, the minimum of mB-sV is 6, and the noise emission will be limited.
Remark :
The case that is to be strictly avoided is when mB-sV can be nul, which causes the order of the Bessel function to be 0. As a consequence, we have to take care having B and V prime numbers.

## Determination of source levels

The minimization of the transfer function ${{F_{m}} \over {P_{m}}}$ is a great step in the process of reducing the noise emission. Nevertheless, to be highly efficient, we also have to predict the source levels $F_{m}$. This will lead us to choose to minimize the Bessel functions for the most significant values of m. For example, if the source level for m=1 is very higher than for m=2, we will not consider the Bessel functions of order 2B-sV. The determination of the source levels is given by the Sears theory, which will not be discussed here.

## Directivity

All this study was made for a specific direction : the axis of the Rotor/Stator. All the results are acceptable when the noise reduction is ought to be in this direction. In the case where the noise to reduce is perpendicular to the axis, the results are very different, as those figures shown :
For B=13 and V=13, which is the worst case, we see that the sound level is very high on the axis (for $\theta =0$)
For B=13 and V=19, the sound level is very low on the axis but high perpendicularly to the axis (for $\theta =Pi/2$)

## Further reading

This module discusses rotor/stator interaction, the predominant part of the noise emission of turbomachinery. See Acoustics/Noise from Cooling Fans for a discussion of other noise sources.

## External references

* Prediction of rotor wake-stator interaction noise by P. Sijtsma and J.B.H.M. Schulten
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