Proceedings of the Institute of Acoustics

Experiments on the aeroacoustic whistling of a cylindrical cavity

Abel Faure-Beaulieu¹, ETH Zürich, Zürich, Switzerland

Tiemo Pedergnana, ETH Zürich, Zürich, Switzerland

Yuan Xiong, Beihang University, Beijing, China

Nicolas Noiray², ETH Zürich, Zürich, Switzerland

ABSTRACT

An air flow going through an axisymmetric cavity can interact with the acoustic modes of this cavity and produce loud self-sustained acoustic oscillations. Acoustic measurements reveal that the nature of the acoustic mode undergoes complex dynamics. Simultaneously to acoustic measurements, time resolved stereoscopic particle image velocimetry gives the three velocity components in a two-dimensional plane. Although the experimental velocity fields are only two-dimensional, the whole tridimensional phase-averaged velocity field can be reconstructed for the specific case of a spinning aeroacoustic mode. This reconstruction reveals the existence of a helical vortex tube spiralling in the shear layer at the cavity opening.

1. INTRODUCTION

Aeroacoustic whistling of deep cavities under a grazing flow is a phenomenon studied since decades because it is a source of unwanted noise and vibration in aerodynamic applications and piping systems [1, 2]. It happens when the hydrodynamic fluctuations of the shear layer at the cavity’s mouth efficiently provide energy to one of the acoustic modes of the cavity, causing acoustic oscillations that amplify in return the hydrodynamic fluctuations of the shear layer. This feedback loop mechanism can produce loud tonal whistling. In deep rectangular cavities, which are a typical academic configuration producing aeroacoustic whistling, the acoustic mode involved in the instability is generally a longitudinal mode of the cavity [3]. In the present study, we focus on a different configuration: an axisymmetric deep cavity. Previous studies on whistling in deep or

Figure 1: Experimental setup. (a) Picture of the cavity. (b) Side cut of the cavity and the duct. (c) Axial view of the cavity and the stereoscopic PIV system (cameras and laser sheet). The three coloured points correspond to the microphones positions.

Shallow axisymmetric cavities have shown that the unstable acoustic modes are azimuthal modes, also sometimes referred to as "diametral modes" [4–7]. Azimuthal modes arise as pairs of degenerate or quasi-degenerate acoustic modes, which leads to more complex dynamics than in the case of the rectangular cavities. The dynamics of azimuthal modes is already a topic of investigation in the domain of gas turbines for power generation and aircraft propulsion, because they are important to model the thermoacoustic instabilities occurring in the annular combustion chambers of these turbines [8–12]. Azimuthal thermoacoustic modes are classified in three categories: spinning modes, propagating in the azimuthal direction at the speed of sound, standing modes, keeping a fixed or slowly evolving nodal line, and mixed modes, being a linear combination of both [9, 13]. The same classification can be applied to aeroacoustic azimuthal modes in whistling axisymmetric cavities. The study of Abdelmwgoud et al. [6] is a compressible LES of the whistling of a shallow axisymmetric cavity. It sheds light on the structure of the shear layer oscillations depending on the nature of the acoustic mode. Spinning modes are associated with a continuous helical vortex tube advected downstream by the grazing flow, while standing modes are characterised by disconnected vortex crescents alternatively produced on both sides of the nodal line of the acoustic mode. However, few experimental studies investigated in depth the hydrodynamic fluctuations of the shear layer. The PIV study from Oshkai and Barannyk [5] captures 2D fields of the vortex shedding associated to the first and second shear layer modes, featuring respectively one and two vortical structures in the width of the cavity opening. In the present paper, we applied for the first time stereoscopic particle image velocimetry to a deep cylindrical cavity subject to aeroacoustic whistling. Stereoscopic PIV allows us to obtain the three components of velocity in a 2D section of the cavity. With the experimental data obtained, we propose a 3D reconstruction of the phase averaged velocity field. This allows to visualize for the first time an experimental reconstruction of the helical vortex tube observed in previous simulations [6, 7]. In a first section, the experimental setup is described. Then, the main features of the instability are commented. The last section describes the process of phase averaging and 3D reconstruction.

2. EXPERIMENTS

The axisymmetric cavity, represented in fig. 1, is a cylinder of radius 128 mm and width 30 mm, with a glass window to allow flow visualisation. The grazing flow is supplied through a cylindrical duct of radius 20 mm, ending with anechoic terminations to avoid coupling of the instability with pipe modes. The air mass flow is controlled with a valve and measured with a Bronkhorst mass flow

Figure 2: Experimental results. (a) Spectrum measured by a microphone placed on the cavity, for different velocities U_x of the grazing flow. (b) Instantaneous snapshot of the three components of the velocity field in the cavity width, captured with the PIV, during an aeroacoustic instability. The bulk velocity is U_x = 47 m/s.meter.

Acoustic measurements are done with three microphones placed on the downstream wall of the cavity at equal radial position 90 mm and different azimuthal locations 90^◦, 208^◦, 332^◦, as shown in fig. 1 (c).

The instantaneous fluctuations of the three velocity components are obtained in a plane section of the cavity with fast stereoscopic particle image velocimetry (PIV). Small DEHS seeding particles are injected into the grazing flow. They are illuminated by a fast pulsed laser sheet. The particle images are taken by two fast cameras placed at 45° of the laser sheet and with orthogonal focal directions, as represented in 1 (c). The imaging acquisition frequency is 6 kHz. The PIV window is shown in fig. 1 (b). Its width is limited by the presence of laser reflections on the side walls of the cavity, and its height is limited by the image distortion due to the curvature of the cylindrical glass window.

3. EXPERIMENTAL RESULTS

An aeroacoustic instability involving the first order azimuthal acoustic mode is observed when the bulk axial velocity U_xin the duct ranges between 42 and 72 m/s. Figure 2 (a) represents the acoustic spectra recorded for different bulk velocity. For U_x = 36 m/s, the spectrum corresponds to broadband noise with small peaks at the acoustic resonances of the cavity. At higher velocities, the instability occurs and a sharp peak arises at 790 Hz, corresponding to the frequency of the first acoustic mode of the cavity, obtained with a Helmholtz solver. A secondary peak is also visible at the harmonic frequency. In the most unstable conditions, the fundamental peak reaches 165 dB.

PIV acquisition was done for several different values of the bulk velocity. Each PIV acquisition covers 0.1 s. The acquisition frequency of 6 kHz is sufficient to resolve well the hydrodynamic fluctuations at the fundamental acoustic frequency (790 Hz). Figure 2 (b) is a snapshot taken with the PIV for U_x = 47 m/s. The axial component u_xis large in the centre, and close to zero in the cavity depth. A cylindrical shear layer (in green) arises from this mean velocity difference. The vertical and out-of plane velocity fields (resp. u_y and u_z) reveal the advection of coherent vortical structures in the shear layer.

Figure 3 shows the acoustic pressure time traces for this same condition U_x = 47 m/s, obtained from the three microphones. Figure 3 (a) reveals that, while the overall amplitude of the acoustic mode remains around 1000 Pa, strong amplitude variations are seen in the individual signals of each microphone. For instance, the blue signal (microphone places 332^◦) is sometimes almost zero, characterizing a standing mode whose nodal line coincides with the position of this microphone. The strong changes of the relative local amplitudes observed here are reminiscent of the phenomenon of

Figure 3: Acoustic pressure timetraces measured with the three microphones for the condition U_x = 47 m/s. The colour code corresponds to the three microphones shown in fig. 1 (c). (a) 2.5 s of acoustic time traces. The vertical dashed lines correspond to the time interval where the PIV images have been acquired. (b) Detail of the PIV time interval. The shaded zone corresponds to a time window where the three signals have almost equal amplitudes, corresponding to a spinning mode. (c) Zoom on the few first oscillations of the PIV time interval.

Beating azimuthal mode already described in the case of thermoacoustic azimuthal modes of annular combustion chambers [14], where the mode alternatively spins clockwise and counterclockwise, with periodic reversals of the propagation direction. However, the time trace shown in fig. 3 (a) are not constantly showing this beating. For instance, between 39.5 and 40.7 s, the amplitudes do not fluctuate strongly. Figure 3 (b) shows the acoustic time traces during the PIV acquisition. In the first half, the three signals have almost equal amplitudes, corresponding to a quasi pure spinning mode. Then, the amplitude measured at position 208^◦ becomes larger, indicating a mixed mode. Figure 3 (c) shows the detail of the oscillations when the mode is quasi pure spinning. The oscillations look perfectly sinusoidal, which is not surprising because the fundamental peak in fig. 2 (a) largely dominates the harmonics and the background turbulent noise. The order of the crests indicates that the wave travels in clockwise direction.

4. TRIDIMENSIONAL RECONSTRUCTION OF THE PHASE AVERAGED VELOCITY FLUCTUATIONS

Figure 4: Phase averaged axial velocity snapshots, for U_x = 47 m/s.

We apply the classical triple decomposition to the velocity field, as introduced by Reynolds and Hussain [15] to describe oscillations in turbulent flows with periodic coherent structures:

Where x is the position vector, u is the tridimensional velocity field, ū is the mean flow, ũ represents the coherent periodic fluctuations associated to the instability, and u'  includes all the non-coherent turbulent fluctuations. We define the phase averaged flow as <u> = ū + ũ, which is the velocity field freed from its stochastic turbulent oscillations. This phase averaged flow can be obtained from the PIV measurements, provided that the number of coherent oscillations captured in the PIV time interval is sufficiently large. As in [3], the phase averaging is done by grouping the PIV images in twelve bins, depending on their phase in the hydrodynamic cycle, and then taking the average of the content of each bin. The phase is defined by extracting a reference velocity signal at one location of the shear layer, where the oscillations are the strongest. This oscillating signal is filtered around its dominant frequency, which is equal to the acoustic frequency. Then, a Hilbert transform allows to obtain the phase signal. Figure 4 displays six of the twelve phase-averaged snapshots of the axial component of <u> at different phases of the aeroacoustic cycle, for the operating condition U_x = 47 m/s, using the PIV data obtained during the shaded time interval in fig. 3 (b), corresponding to a situation where the mode spins clockwise. It clearly shows the perturbations advected in the shear layer.

We now propose to use this phase averaged series of snapshots to reconstruct the tridimensional phase averaged velocity field in the shear layer. This is not possible in general because the PIV gives only 2D information. However, in the specific case of spinning modes, a translation in time is equivalent to a rotation around the axis. In the case of a clockwise spinning mode, the phase indicated on top of each snapshot of 4 can be interpreted either as a temporal phase or as an angular position. With this second interpretation, the snapshots correspond to slices of the velocity field at different equispaced azimuthal angular positions. Then, the whole 3D phase-averaged velocity field is obtained by interpolating the velocity field between the slices. Figure 5 (a) represents a positive and a negative isosurface of the radial coherent velocity fluctuations, revealing a helical structure in the shear layer. Figure 5 (b) is a Q

Figure 5: (a) 3D reconstruction of the radial velocity fluctuations ũ_r The green and blue isosurfaces corresponds respectively to +5 m/s and -5 m/s. (b) Isosurface of Q criterion. The blue arrow is the spinning direction of the aeroacoustic mode. The bulk velocity is U_x = 47 m/s

Isosurface showing a helical vortex tube spiralling in the shear layer, and whose winding direction is opposite to the spinning direction of the wave. This is explained by the fact that vortical perturbations continuously detach in a clockwise spinning motion from the upstream edge of the circular cavity opening under the action of the spinning acoustic wave, before being convected downstream in the axial direction.

5. CONCLUSIONS

The aeroacoustic whistling of an axisymmetric deep cavity was investigated with simultaneous acoustic and stereoscopic PIV measurements. Phase averaging is applied on the PIV sequence to cancel the non-coherent turbulent fluctuations and obtain an average hydrodynamic cycle. Acoustic measurements allow to identify the behaviour of the mode and to select a time interval where the mode is spinning, for which full 3D reconstruction of the coherent hydrodynamic oscillations is possible. The hydrodynamic structure involved in spinning aeroacoustic modes is a continuous helical vortex tube, in agreement with the LES results from [6, 7]. This helical shape comes from the simultaneous detachment of perturbations from the edge of the cavity opening and their further advection by the mean flow.

ACKNOWLEDGEMENTS

This project has received funding from the European Union’s Horizon 2020 research and innovation programme under Grant Agreement No. 765998.

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¹abelf@ethz.ch

²noirayn@ethz.ch