The analysis of aeroacoustic phenomena is crucial for a deeper understanding of the damping mechanisms of a sound-absorbing bias flow liner (BFL). For this purpose, simultaneous measurements of the sound field and the flow field in a BFL are required. The fluid velocity can serve as the measurand, where both the acoustic particle velocity and the aerodynamic flow velocity contribute and, thus, can be acquired simultaneously. However, there is a need to separate these two quantities to distinguish between them. This is challenging because they generally coincide with each other in the time domain. Due to the interaction of sound and flow in a BFL, both velocities also overlap in the temporal frequency domain, having a coherent oscillation at the acoustic frequency. For this reason, the recently developed natural Helmholtz–Hodge decomposition (NHHD) is applied to separate both quantities from the measured oscillation velocity field in the spatial domain. The evaluation of synthetic vector field data shows that the quality of the decomposition is enhanced when a smaller grid size is chosen. The velocity field in a generic BFL, necessarily recorded within a three-dimensional region of interest at more than 4000 measurement locations, is evaluated using NHHD. As a result, the measured oscillation velocity in the BFL is dominated by the flow that is related to vortices and also by irrotational aerodynamic flow. Moreover, indications for an aeroacoustic source near the facing sheet of the liner are revealed.

In order to improve the efficiency of modern aeroacoustic noise absorbers in
aircraft engines, a deeper understanding of the underlying damping phenomena
is necessary, according to

Previous separation approaches using the Proper Orthogonal Decomposition
(POD) of the measured velocity field, like in

To split a spatially bounded vector field into its solenoidal and its
irrotational part, the conventional HHD has previously been applied in many
research fields, including fluid mechanics, by

In the following paragraphs, the theoretical aspects of the NHHD from

According to the ideas of

Performing the NHHD means calculating the divergent term

The NHHD algorithm from Sect.

In order to verify the implementation from Sect.

Synthetic input vector field terms

A synthetic two-dimensional vector field with a quantity of dimension one is
selected as input data for the NHHD algorithm. The original (Euclidean)
vector field

For comparison with the output vector field terms in the following
subsection, the divergence and curl of the input vector field terms are
calculated numerically using central finite differences; cf.
Eq. (

Output vector fields

Using its implementation according to Sect.

Resulting crosstalk, according to Eq. (

The crosstalk from Eq. (

In order to obtain experimental data for the aeroacoustic analysis, an
optical measurement of the velocity vector field in a generic bias flow liner
is performed using frequency-modulated Doppler global velocimetry (FM-DGV)
from

The aeroacoustic experiment in a generic bias flow liner is conducted in
Berlin at the “Duct acoustic test rig with rectangular cross section”
(DUCT-R), which is about 3

Top view of the setup of the aeroacoustic measurement in a generic bias flow liner with optical access through glass windows; the point of origin is at the center of the central orifice.

Output vector fields (

The design of the bias flow liner is based on previous work by

The velocity data are measured by FM-DGV at eight locations in parallel,
linearly arranged in

Root mean square (rms) of the output vector fields (

The NHHD is applied to the three-dimensional vector field of the measured
fluid velocity in order to separate the acoustic particle velocity and the
aerodynamic flow velocity. Here, this is only done for the oscillation
velocity at the acoustic excitation frequency

The phase-averaged oscillation velocity vector field, denoted by

In order to provide further insight, the NHHD is applied to the measured
data following Sect.

The irrotational term

The solenoidal term

The harmonic term

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The aim of this work is to perform an aeroacoustic analysis of an
experimentally obtained volumetric velocity vector field, measured in a bias
flow liner. For this purpose, the oscillation velocity with respect to the
frequency of the sinusoidal acoustic excitation is evaluated using the
natural Helmholtz–Hodge decomposition (NHHD). This vector field
decomposition of the fluid velocity into the irrotational acoustic particle
velocity and the solenoidal flow velocity is successfully applied on
a three-dimensional vector field in a 1

The analysis shows that the flow velocity, which is related to vortices,
dominates in the fluid velocity field, which coincides with previous
experimental work by

Furthermore, the influence of the grid size on the quality of the NHHD is investigated using synthetic data. As a result, the grid size has to be minimized in order to minimize computational errors. However, this yields a higher computational effort.

In the future, the employment of parallel computing, e.g., using a graphics
processing unit, should be considered to save calculation time (see

Due to their large volume (16 TB), the underlying data can not be accessed online; however the data are stored on local mass storage devices at TU Dresden. Please contact the corresponding author regarding any data access.

The authors declare that they have no conflict of interest.

The authors thank the German Research Foundation (DFG) for the sponsoring of the projects CZ 55/25-3 and EN 797/2-3. Many thanks go to Toptica Photonics for continuous support concerning the modulatable, high-power diode laser. Harsh Bhatia from the Lawrence Livermore National Laboratory is gratefully acknowledged for fruitful discussions. Further thanks are due to Heiko Scholz for supporting the implementation in MATLAB. Edited by: Rainer Tutsch Reviewed by: two anonymous referees