Characterization of the aeroacoustic instability in a T-junction
Claire BOURQUARD1 , Abel FAURE-BEAULIEU1 , Nicolas NOIRAY1
1 CAPS Laboratory, Department of Mechanical and Process Engineering, ETH Zürich, 8092 Zürich, Switzerland

Abstract
Aeroacoustic resonance at a T-junction is a well-known phenomenon that has been studied using a variety ofdifferent approaches. These include vortex-sheet modelling of the shear layer and Linearised Navier-Stokes
Equations (LNSE) incorporating a turbulence model. The aeroacoustic instability results from the constructive
interaction of the acoustic waves in the cavity and the turbulent shear layer. In the present study, measurementsof the acoustic impedance of the shear layer and of the cavity are performed at different frequencies and mean
flow velocities. These measurements are used to derive a linear model of the aeroacoustic instability and allow
for analysis of linear stability.
Keywords: Aeroacoustics, Linear model, Impedance

1INTRODUCTION

Flow past the opening of a closed side-branch, as for example performed in a T-junction, is of great concern ina wide variety of practical applications such as ducts, pipe and ventilation systems or even wind instruments,
since the coupling between the acoustic waves in the closed branch and the turbulent shear layer can result inaeroacoustic resonance. This phenomenon has been the subject of multiple studies since more than 50 years,
beginning with wind tunnel experiments and first empirical models [8, 1, 10]. In later studies, more detailed
modelling of the shear layer was performed by using vortex-sheet theory [7, 5]. Latest studies include numericalsimulations to discuss the influence of the cavity edges on the generation mechanism [4], thorough experimental
investigations [6] and instability modelling using Linearised Navier-Stokes Equations (LNSE) incorporating aturbulence model [2]. The present study aims at bridging a simple analytical model of two coupled oscillatorswith the experiments performed on a rectangular T-junction.
In a first part, the analytical model used for modelling the T-junction aeroacoustic resonance will be derived.
In a second part, the experimental campaign will be described. In the last part, the results of the model withfitted parameters will be compared with the full T-junction aeroacoustic experiments.

2ANALYTICAL MODEL

2.1 Acoustic-hydrodynamic coupled system
The configuration considered in the present study consists in a turbulent air flow of mean velocity U running
through a channel of square cross section with a perpendicular side branch of rectangular cross section, andwhose depth L can be manually adjusted with a piston. A sketch of the system can be seen in Fig. 1.
Since the fluid in the side-branch is quiescent, a turbulent shear layer forms between the main channel and theside branch. Depending on the mean flow velocity U and the side-branch length L, constructive feedback canbe achieved between the acoustic waves travelling inside the side branch and the shear layer, resulting in an
aeroacoustic instability. For the model, the system will be divided in two coupled subsystems: the side branchon the one side, and the shear layer on the other side, coupled at the reference plane y = 0− . The side branch
is modelled as a damped acoustic oscillator, following the quarter-wave resonator modelling of [3]. We noteũ the acoustic velocity in m/s and u = ρcũ the "normalized" acoustic velocity in Pa. With this, the specificadmittance of the side branch at reference plane y = 0− can be expressed in the frequency domain as follows:

1 clairebo@ethz.ch, noirayn@ethz.ch2120

Anechoic endx
Uy

Side branch
Eq. (1)
Microphonesx

Ly=+ ShearLayer
Eq. (2)

UUFigure 1. Sketch of the resonant T-junction system, divided into separate acoustical elements : side branch (Eq.
1) and shear layer (Eq. 2). Please note that the shear layer element also includes the area expansion, whichmeans that the response of this element will be a mix of acoustic and hydrodynamic response.

AP (s) =γsû,=− 2p̂
s + 2αs + ωa2(1)

with p the acoustic pressure, s = iω the Laplace variable, and ω is the varying angular frequency. α is thedamping, γ represents the inverse of the effective mass of the oscillator, and ωa is the resonance angularfrequency of the side branch. The value of the parameters α, γ, and ωa depends on the side branch lengthL. The subscript "a" stands for "acoustic", since we will label the side branch subsystem as the "acousticoscillator". A higher level of complexity was needed to capture the acoustic response of the shear layer, which
is why in this case a transfer function of order 2 was used for the specific impedance as follows:
ZSL (s) =s2 + 2ds + ω02p̂
=G 2,ûs + 2bs + ωh2(2)with G, b, d, and frequencies ω0 and ωh depending on the mean flow velocity U and to be determined from thefit in the next section. In reality, this subsystem entails the shear layer as well as the expansion from the side
branch to the channel. For this reason, it is a mix of acoustic and hydrodynamic response. However, for thesake of simplicity, this subsystem will be labelled as the "hydrodynamic oscillator", which is why the subscript"h" is used, standing for "hydrodynamic". Using the two previously defined subsystems, the coupled system intime domain becomes: ü + 2α u̇ + ωa2 u = −γ ṗ(3)p̈ + 2b ṗ + ωh2 p = G(ü + 2d u̇ + ω02 u),Using the first equation to express the second derivative of the acoustic velocity ü, the system can be rewritten:
 ü + 2α u̇ + ωa2 u = −γ ṗ(4)p̈ + 2β ṗ + ωh2 p = µ u̇ + λ u,2121

With β = (2b + Gγ)/2, µ = 2G(d − α) and λ = G(ω02 − ωa2 ). The original system was now reduced to a simple
coupled system of damped harmonic oscillators with a mix of dissipative and reactive coupling.
2.2 Stability u
The system can be written in matrix form using x =:p 2
1 02αγ
ω0ẍ +ẋ + ax = 0.
0 1−µ 2β−λ ωh2| {z }| {z }| {z }
MD(5)KThus, the stability of this linear system depends on the sign of is eigenvalues, which are the roots of:det(s2 M + sG + K) = (s2 + 2αs + ωa2 )(s2 + 2β s + ωh2 ) + γs(µs + λ ) = 0.

(6)

The characteristic polynomial of the system is:
s4 + s3 2(α + β ) + s2 (ωa2 + ωh2 + 4αβ + γ µ) + s(2αωh2 + 2β ωa2 + γλ ) + ωa2 ωh2 .

(7)

The roots of this polynomial correspond to the eigenvalues of the coupled system, which will be compared tothe spectrum of the aeroacoustic instability for different velocities in Section 4.

3EXPERIMENTS

3.1 Aeroacoustic resonance
The system considered in the present study consists of a 62x62 mm2 channel about 2m long, crossed by a flowof mean velocity U varied between 35 and 75 m/s. In the middle of the channel, a side branch of length L(varied between 0 and 380mm with a piston) and of cross section 62x30mm2 is connected to the main channel
via a sharp-edged perpendicular T-junction. For the rest of the study, we fix the side branch length at L = 250
mm. As can be seen in Fig. 1, four G.R.A.S. 46BD 1/4" CCP microphones are installed inside the side branch.
The pressure at the fourth microphone, at the end of the side branch, is measured for different velocities,
varying from 35 to 75 m/s. The results in terms of spectrum is shown in Fig. 2.

Figure 2. Acoustic mapping of the shear layer - side branch T-junction aeroacoustic resonance in terms ofPower Spectral Density (PSD) for different mean flow velocities U for L = 250mm. The spectra for chosenvelocities (colored dashed lines) will be compared to the fitted model in Section 4.

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Loudspeakers

Microphones
x4x3

x2 x10

Acoustical element
Anechoic end

Figure 3. Sketch of the impedance tube used for the subsystems reflection coefficient measurements
3.2 Identification of the model parameters
For the determination of the parameters in Eqs. 1 and 2, the acoustic response of the side branch on the one
side, and of the shear layer on the other side, is measured. For this purpose, reflection coefficient measurements
are performed using the multi-microphone method (MMM) [9] in an impedance tube of 62x30 mm2 crosssection, equipped with 2 Beyma CP850Nd compression drivers facing each other, and 4 (for the side branch) or
6 (for the shear layer) G.R.A.S. 46BD 1/4" CCP microphones. On the microphone side end of the impedance
tube, the acoustic element to be measured is placed (in the present case, either the side branch or the expansion+ shear layer from Fig. 1). On the loudspeaker side end of the impedance tube, an anechoic end is installed,which prevents acoustic waves reflection. A sketch of the experimental setup can be seen in Fig. 3.
Once the reflection coefficient has been measured, the specific impedance can be obtained using Z = (R −1)/(R + 1), and the specific admittance A = 1/Z. Following this, the measurements can be fitted using Eqs. 1
and 2. The results of the measurements and fits can be found in Fig. 4.

Figure 4. (a) and (b): modulus and phase of the specific admittance of the side branch for L = 250 mm from
MMM measurements. The black dashed line corresponds to the fit based on Eq. 1. (c) and (d): real andimaginary part of the shear layer impedance for low excitation amplitude and different velocities U. The black
dashed lines correspond to the fit based on Eq. 2. Please note the range of frequencies for which ℜ(Z) < 0,which means that the shear layer amplifies disturbances in this region and might lead to an instability.

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Figure 5. Most unstable eigenvalue of the coupled system for varying mean flow velocity U through the channel(top). Spectra of the aeroacoustic coupled system for the same velocities (bottom), corresponding to the coloreddashed lines in Fig. 2.

4COMPARISON OF THE EXPERIMENTS WITH THE FITTED MODEL

The eigenvalues of the fitted model can now be compared with the frequency response of the experimental
coupled system. This comparison between the most unstable eigenvalue of the fitted model (i.e. the one withthe highest real part) and the experimental spectra from Fig. 2 is shown in Fig. 5. The imaginary part ofthe model eigenvalues corresponds to the frequency of the mode, which is in excellent agreement with thefrequency of the highest peak in the spectra. For U = 74.1 and U = 70.6 m/s, the modelled system has aneigenvalue whose real part is positive, meaning that the system is unstable. In the experimental power spectral
density (PSD), the instability is characterized by a sharp peak, which underlines the good agreements between
fitted model and experiments in terms of stability.

5CONCLUSIONSIn the present study, the aeroacoustic resonance of a T-junction coupling a shear layer and a side branch wasstudied. For this purpose, the frequency response of the coupled system was measured at different mean flow
velocities U. Then, the acoustic admittance and impedance of the two subsystems (side branch and shear layer)
were measured separately by the multi-microphone method. An analytical model was derived for the admittanceof the side branch and for the impedance of the shear layer, which parameters could be fitted on the previousmeasurements. The eigenvalues of the fitted coupled system were in very good agreement with the experimental
spectra of the aeroacoustic instability, both in terms of frequency and stability.

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