On impedance conditions for circular multiperforated acoustic liners
Abstract
Background
The acoustic damping in gas turbines and aero-engines relies to a great extent on acoustic liners that consists of a cavity and a perforated face sheet. The prediction of the impedance of the liners by direct numerical simulation is nowadays not feasible due to the hundreds to thousands repetitions of tiny holes. We introduce a procedure to numerically obtain the Rayleigh conductivity for acoustic liners for viscous gases at rest, and with it define the acoustic impedance of the perforated sheet.
Results
The proposed method decouples the effects that are dominant on different scales: (a) viscous and incompressible flow at the scale of one hole, (b) inviscid and incompressible flow at the scale of the hole pattern, and (c) inviscid and compressible flow at the scale of the wave-length. With the method of matched asymptotic expansions we couple the different scales and eventually obtain effective impedance conditions on the macroscopic scale. For this the effective Rayleigh conductivity results by numerical solution of an instationary Stokes problem in frequency domain around one hole with prescribed pressure at infinite distance to the aperture. It depends on hole shape, frequency, mean density and viscosity divided by the area of the periodicity cell. This enables us to estimate dissipation losses and transmission properties, that we compare with acoustic measurements in a duct acoustic test rig with a circular cross-section by the German Aerospace Center in Berlin.
Conclusions
A precise and reasonable definition of an effective Rayleigh conductivity at the scale of one hole is proposed and impedance conditions for the macroscopic pressure or velocity are derived in a systematic procedure. The comparison with experiments show that the derived impedance conditions give a good prediction of the dissipation losses.
Keywords
Acoustic liner Perforated plates Multiscale analysis Rayleigh conductivity Impedance conditionsMSC
35Q30 35B27 74Q15 76M501 Introduction
The safe and stable operation of modern low-emission gas turbines and aero-engines crucially depends on the acoustic damping capability of the combustion system components. Hereby, so called bias flow liners—consisting of a cavity and a perforated face sheet with additional cooling air flow—play a significant role. Since decades the damping performance prediction of these bias flow liners under all possible flow conditions remains a major challenge. However, due to the higher tendency of low-emission, lean burn combustion concepts for combustion instabilities the prediction of the acoustic bias flow liner impedance and therewith its damping performance is a very important prerequisite for the engine design process. Several analytical and semi-empirical models for the impedance description of bias flow liners were developed in the past (see also [1]). This work focuses on the numerical simulation of the acoustic characteristics of bias flow liners applying multi-scale modeling.
Indeed, the (effective) Rayleigh conductivity depends on the geometrical parameters, especially size and shape of the holes and their distances as well as the physical parameters, noting at first the viscosity and the frequency. Some of these parameters take small values and we consider them to be scaled with a small parameter \(\delta> 0\) in such a way that making δ smaller the effective Rayleigh conductivity stays essentially constant and tends in the limit \(\delta\to0\) to a non-trivial value.
2 Methods
We consider an acoustic liner that consist of a wall or part of a wall with a periodic dense array of equisized and equishaped holes with a characteristic periodicity that is proportional to the small parameter δ. The holes may not be of cylindrical shape and even tilted in general. For sake of simplicity we consider the perforated wall \(\varOmega_{\mathrm {liner}}^{\delta}\) with a circular cross-section of fixed inner radius \(R_{\mathrm {d}}\), while noting that the proposed procedure to define the Rayleigh conductivity and impedance conditions do not depend on the choice of the cross-section, but only on the hole pattern and hole shape and can be directly transfered to other cross-sections like rectangular.
In the following section we study the solution of the viscoacoustic model in three different geometrical scales beginning at the scale of one hole, pursuing with the scale of one period of the hole array and concluding with the macroscopic scale, on which the impedance conditions follow.
2.1 Microscopic scale: the near field around one hole
Note that in problem (6b)–(6d) the term \(-\nu_{0}' \nabla \operatorname {div}\tilde{\boldsymbol {\mathfrak {v}}}\) that would appear in the first line cancels out due to the divergence free condition (6b). Moreover, note that the term \(-\nu_{0} \Delta\tilde{\boldsymbol {\mathfrak {v}}}\) can be replaced by \(\nu_{0} \,\mathbf{curl}\,\mathbf{curl}\,\tilde {\boldsymbol {\mathfrak {v}}}\) and so only the vorticity part of the velocity \(\tilde{\boldsymbol {\mathfrak {v}}}\) will exhibit a viscosity boundary layer as we will see later.
Problem (6b)–(6d) is a classical saddle-point problem and admits a unique solution stated by the following proposition.
Proposition 2.1
There exists a unique solution\((\tilde{\boldsymbol {\mathfrak {v}}},\tilde{\mathfrak {p}}) \in(\mathrm {H}^{1}(\widehat{\varOmega}))^{3} \times \boldsymbol {\mathcal {V}}(\widehat{\varOmega})\)of (6b)–(6d), where\(\boldsymbol {\mathcal {V}}(\widehat{\varOmega}) = \lbrace P \in \mathrm {H}_{\mathrm {loc}}^{1}(\widehat{\varOmega}) \textit{ such that } \|\nabla P\|_{\mathrm {L}^{2}(\widehat{\varOmega})} < \infty \rbrace\).
Note, that the pressure space \(\boldsymbol {\mathcal {V}}(\widehat{\varOmega})\) allows for a constant behavior towards infinity.
Note, that the normal component of the near field velocity profile \(\boldsymbol {\mathfrak {v}}\) decays like \(1/S^{2}\) towards infinity and combines different behaviour close to and away from the wall (see Fig. 3(c) and (d)). This behaviour can be rigorously justified with similar techniques as in [6, 7].
For the usual definition of the Rayleigh conductivity \(K_{R}\) it is not evident where the difference of the pressure—as it varies locally—and the volume flux—as in the original acoustic equations the fluid is compressible—shall be evaluated. The quantity \(k_{R}\) is, however, clearly defined by (6b)–(6d) and (8) as the near field pressure tends to constant values for \(|\mathfrak {X}| \to\infty\) and as the near field velocity is incompressible. This results from the separation of the effects at the different length scales, namely viscous incompressible behaviour in the vicinity of the holes versus inviscid, compressible behaviour away from them, due to the asymptotic ansatz. As the near field profiles are defined in local coordinates \(\mathfrak {X}\) it has the dimensions of one over length and we denote it as effective Rayleigh conductivity of the liner.
The definition of the effective Rayleigh conductivity \(k_{R}\) can be used for inviscid fluids as well, for which \(\nu_{0} = 0\), if the no-slip boundary conditions (6c) are replaced by \(\boldsymbol {\mathfrak {v}}\cdot \mathbf {n}= 0\).
2.2 Mesoscopic scale: the hole pattern
Note that (11) is equivalent to an homogeneous Laplace problem with Neumann boundary conditions for the pressure profile \(\mathcal {P}^{\delta}\), where the velocity profile \(\boldsymbol {\mathcal {V}}^{\delta}\) can be computed from. Following [8, Proposition 2.2], we can therefore state the following proposition.
Proposition 2.2
2.3 Macroscopic scale and impedance conditions
Distinguished limit
Note, that the nature of the impedance condition (19b) is due to the choice of asymptotic scales. It represents a distinguished limit meaning that different choice would lead to one of the trivial conditions \([p_{0}](\mathbf {x}_{\varGamma}) = 0\) (transparent wall) or \(\langle \mathbf {v}_{0}\cdot \mathbf {n}\rangle(\mathbf {x}_{\varGamma}) = 0\) (rigid wall), as it was already stated in [12, Eq. (4.4) and below] for infinitely thin perforated wall and the Stokes flows. If we would scale the diameter of each hole with \(\varepsilon (\delta)\) as well as the thickness of the perforated wall such that \(\delta^{2} = o(\varepsilon (\delta))\) then we would obtain transparent wall conditions in the limit \(\delta\to0\), e.g., if the diameter of each hole scales like δ. A contrario, the impedance conditions become rigid wall conditions if we would use the scaling \(\varepsilon (\delta) = o(\delta^{2})\).
Acoustic impedance
Formulation in pressure only
3 Results and discussion
In this section, we are interested by the numerical computation of the effective Rayleigh conductivity \(k_{R}\), the computation of dissipation losses in acoustic ducts with the impedance conditions and comparison with data from experimental measurements.
3.1 Numerical computation of \(k_{R}\)
The effective Rayleigh conductivity \(k_{R}\) is defined through the solution of the near field velocity and pressure profiles in the unbounded domain Ω̂ around a single hole. To compute \(k_{R}\) numerically we truncate the unbounded domain, on which we use the finite element method for discretization and propose an extrapolation procedure to increase the accuracy.
As, more precisely, the Rayleigh conductivity \(k_{R}\) can be expanded in powers of \(1/S\) we use an extrapolation in \(1/S\) of first order approximations \(k_{R}(S)\) for different truncation radia S to obtain a second or higher order approximation of the limit value \(k_{R}\).
Liner configurations. The length of the liner is \(L=60~\text{mm}\). The value of the viscosity is \(\nu(\delta) = 1.4660 \times10^{-5}~\text{m}^{2}\mbox{/}\text{s}\). For all these configurations \(b=0.5\sqrt{ \mathrm {a}}\)
Config. | Number of holes (longitudinal, azimuthal) | Longitudinal inter-hole distance \(\delta/ \sqrt{ \mathrm {a}}\) (mm) | Azimuthal inter-hole distance \(\sqrt{ \mathrm {a}} \delta\) (mm) | Hole diameter \(\mathrm {d}_{0} \delta^{2}\) (mm) | Liner thickness \(\mathrm {h}_{0} \delta^{2}\) (mm) | σ % |
---|---|---|---|---|---|---|
DC006 | (7, 52) | 8.5 | 8.45 | 1 | 1 | 1.1 |
DC007 | (3, 20) | 22 | 21.99 | 2.5 | 1 | 1.0 |
DC008 | (7, 52) | 8.5 | 8.45 | 2.5 | 1 | 6.8 |
DC009 | (3, 20) | 22 | 21.99 | 1 | 1 | 0.2 |
For four liner configurations, see Table 1, from experimental studies [1, 18] we have computed the near field velocity and pressure profiles and so the effective Rayleigh conductivity. The relative kinematic viscosity \(\nu_{0}\) is computed as quotient of the kinematic viscosity \(\nu= 1.4660 \times10^{-5}~\text{m}^{2}\mbox{/}\text{s}\) of air at 15^{∘}C divided by the period δ to the power of four. In Fig. 3(b) and Fig. 3(c) we illustrate the near field pressure and velocity profiles \(\tilde{\mathfrak {p}}\) and \(\tilde{\boldsymbol {\mathfrak {v}}}\) for the liner DC006 at frequency \(306~\text{Hz}\) using a truncation radius \(S=40\). It is visible that the pressure decays almost linearly inside the cylindrical hole, but also the behaviour at distance to the hole. Moreover, the pressure shows close to the rim of the cylinder an edge singularity (i.e., a corner singularity for the 2D axis-symmetric problem) that is resolved numerically by the hp-adaptive refinement strategy. The near velocity profile shows a flux from all sides to and through the hole. It appears that the outward flux of the imaginary part of \(\tilde{\boldsymbol {\mathfrak {v}}}\) over \(\varGamma_{+}(S)\) is negative (resp. positive over \(\varGamma_{-}(S)\)) corresponding to a positive real part of the approximate Rayleigh conductivity \(k_{R}(S)\) (see (24)) and so of the Rayleigh conductivity \(k_{R}\). This is in line with the inviscid case, where \(k_{R}\) is real and positive. Moreover, we see the higher velocity amplitude inside the hole that decays towards its boundaries. This boundary layer phenomena is more visible for lower frequencies (see Fig. 3(d)), where one also see a local change of the velocity direction on the wall boundary.
3.2 Dissipation losses in acoustic ducts
3.2.1 Experimental setup and analysis
The test duct consists of two symmetric measurement sections (section 1 and section 2 in Fig. 8) of 1200 mm length each. They have a circular cross-section with a radius \(R_{\mathrm {d}}\) of 70 mm. In order to minimize the reflection of sound at the end of the duct back into the measuring section the test duct is equipped with anechoic terminations at both ends (not shown in Fig. 8). Their specifications follow the ISO 5136 standard. The damping module is a chamber of 60 mm. It has a circular cross-section with a radius of 120 mm.
A total of 12 microphones are mounted flush with the wall of the test duct. They are installed at different axial positions upstream and downstream of the damping module and are distributed exponentially with a higher density towards the damping module. Two microphones are installed opposite of each other at the same axial position close to the signal source. As evanescent modes become more prominent in the vicinity of the source, their influence is reduced significantly by using the average value of these two microphones for the analysis. This technique helps to reduce the errors for frequencies approaching the cut-on frequency of the first higher order mode and thus, extends the frequency range for accurate results.
At the end of each section a loudspeaker is mounted at the circumference of the duct (A and B in Fig. 8). They deliver the test signal for the damping measurements. The signal used here is a multi-tone sine signal. All tonal components of the signal are in the plane wave range. The signal has been calibrated in a way that the amplitude of each tonal component inside the duct is about 102 dB.
The microphones used in these measurements are 1/4″ G.R.A.S. type 40BP condenser microphones. Their signals are recorded with a 16 track OROS OR36 data acquisition system with a sampling frequency of 8192 Hz. The source signals for the loudspeakers are recorded on the remaining tracks. The test signal is produced by an Agilent 33220A function generator. The signals are fed through a Dynacord L300 amplifier before they power the Monacor KU-516 speakers.
The recorded microphone signals are transformed into the frequency domain using the method presented by Chung [21]. This method rejects uncorrelated noise, e.g., turbulent flow noise, from the coherent sound pressure signals. Therefore, the sound pressure spectrum of one microphone is determined by calculating the cross-spectral densities between three signals, where one signal serves as a phase reference. In our case the phase reference signal is the source signal of the active loudspeaker. As a result we obtain a phase-correlated complex sound pressure spectrum for each microphone signal.
3.2.2 Numerical simulation of dissipation losses
3.3 Numerical results and comparison with experimental data
4 Conclusions
It has been shown that impedance conditions with one numerically computed parameter—the effective Rayleigh conductivity—can predict well the dissipation losses of acoustic liners. The effective Rayleigh conductivity can be obtained by solving numerically an instationary Stokes problem in frequency domain of one hole with a scaled viscosity in an characteristic infinite domain with prescribed pressure at infinity. For the computation the infinite domain is truncated, where we propose approximative boundary conditions on the artificial boundaries and an extrapolation procedure to save computation time. We decoupled in a systematic way the effects at different scales and derived impedance conditions for the macroscopic pressure or velocity based on a proper matching of pressure and velocity at the different scales. In difference to a direct numerical solution for acoustic liners the overall computation effort is separated into a precomputation of the effective Rayleigh conductivity impedance conditions, where no holes have to be resolved anymore by a finite element mesh. The comparison with measurements in the duct acoustic test rig with a circular cross-section at the German Aerospace Center in Berlin show that the dissipation losses based on the impedance conditions with effective Rayleigh conductivity are well predicted. The derivation of the impedance conditions do not depend on the cylindrical shape of the liner and can be used for others shapes like rectangular profiles. The procedure for the computation of the effective Rayleigh conductivity can not only be extented to include thermic effects that are currently only heuristically incorporated, but also nonlinear effects inside the hole that lead to an interaction of frequencies.
Notes
Acknowledgements
The authors would like to thank Claus Lahiri (Rolls-Royce) for fruitful discussions. The research was partly conducted during the stay of the first and second author at the TU Berlin and the first author at BTU Cottbus-Senftenberg.
Availability of data and materials
Not applicable.
Authors’ contributions
The impedance conditions are mainly derived by the first three authors that also conducted the numerical studies and wrote the associated sections of the article. The fourth author contributed mainly with the description of the measurements of the studied liners. All authors participated on the design of the numerical studies and the comparison with the measurements. All authors read and approved the final manuscript.
Funding
The research was supported by Einstein Center for Mathematics Berlin via the research center MATHEON, Mathematics for Key Technologies, in Berlin as well as the Brandenburgische Technische Universität Cottbus-Senftenberg through the Early Career Fellowship of the second author.
Competing interests
The authors declare that they have no competing interests.
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