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Abstract

As a typical noise-attenuating device, Helmholtz resonators are widely implemented in aero-engines and gas turbines to decrease the transmission of acoustic noise. However, an asymmetric Helmholtz resonator could be designed and implemented due to the limited space available in the engines. To examine and optimize the noise-attenuating performances of the asymmetric resonator, comparison studies are performed. For this, a two-dimensional frequency-domain model of a cylindrical duct with a grazing flow is developed. An asymmetric Helmholtz resonator is attached as a side branch. The model containing the linearized Navier–Stokes equations is validated first by comparing the predicted results with the experimental ones available in the literature. Further validation is conducted by comparing the results of an asymmetric resonator with the analytical ones available in the literature. The effects of (1) neck offset distance from the center of the resonator cavity denoted by $e_{c} / a_{s}$ and (2) the grazing flow Mach number $M_{u}$ are evaluated. It is shown that as the grazing flow Mach number is increased, the resonant frequencies and the maximum transmission losses are dramatically varied for a given $e_{c} / a_{s}$ . As $e_{c} / a_{s}$ is increased from 0 to 0.5 and $M_{u} \geq 0.1$ , the resonant frequencies and the maximum transmission losses are increased. However, when $M_{u}$ is lower than 0.07, i.e. $M_{u} \leq 0.07$ , the transmission loss performances are almost unchanged with $e_{c} / a_{s}$ increased. The optimum design of the asymmetric resonator is shown to give rise to the resonant frequency being shifted by 10% and 2–5 dB more transmission loss at higher Mach number. Finally, visualization of vortex shedding formed at the neck of the asymmetric resonator confirms that acoustical energy is transformed into kinetic energy and absorbed by the surrounding air. This study opens up a numerical design approach to optimize an asymmetric resonator.

Keywords

Thermoacoustics Helmholtz resonator acoustic damper transmission loss grazing flow noise damping

Introduction

Undesirable tonal or broadband acoustic noise is present in aero-engines or gas turbine engines.^1–5 To dissipate this noise, Helmholtz resonators are widely implemented in the engine systems.^6,7 In practical engines, a Helmholtz resonator may be attached to a combustor to dampen thermoacoustic instability,⁸ as a grazing flow is present. Such instability is characterized by periodic acoustic pressure perturbations; its amplitude may be up to 5–15% of the engine mean pressure. To mitigate these acoustic fluctuations,^9–12 Helmholtz resonators are designed and implemented. With these resonators implemented, acoustic pulsations may be attenuated somehow and small-amplitude acoustic perturbations are acceptable in engines to increase the fuel–air mixing. A typical Helmholtz resonator¹³ consists of a large cavity, inside which the fluid could be resonated at certain frequencies (also known as resonant frequencies). The resonator and the combustor are connected via a “neck” (a short tube). Acoustic perturbations propagating to the resonator could excite its neck mass—the fluid in the cavity to “resonate.” The resonant frequency is strongly related to the resonator’s geometry and its dimensions (such as the neck and cavity). Thus, such resonance leads to noise absorption being achievable through the effects of thermoviscous (linear) and vortices (nonlinear) generation and shedding.¹⁴ These are the dominant noise-attenuating mechanisms of symmetric¹⁵ or asymmetric Helmholtz resonators.

Intensive studies and researches have been performed on Helmholtz resonators^8,16 with the aim of shedding light on the design, practical implementation, and performance characterization such as power absorption coefficients, acoustic impedance, transmission loss (TL). A nonlinear Helmholtz resonator model (similar to the one proposed by Cummings¹⁷) was derived by Bellucci et al.⁸ to estimate its power absorption coefficient and acoustic impedance, and to characterize the nonlinear vorticity-involved attenuating physics. Validation of this model was done by comparing with the experimental results from an acoustic duct without a grazing flow. Attaching a Helmholtz resonator to an engine leads to a large acoustic impedance at its resonant frequency to acoustic pressure disturbances propagating and thus a great TL, since these sound waves could be completely blocked. However, the resonant frequencies of Helmholtz resonators are distributed over certain frequency bands only. Thus, the resonator’s effectiveness is restricted and limited.¹⁸ To optimize the resonator’s damping performance over a broader frequency range, actively passive control of Helmholtz resonators was recently proposed. This optimization is achievable due to the fact that the resonant frequency depends strongly on its geometric dimensions,⁷ such as the length and cross-sectional area of the neck. In a practical engine system,^19–22 there is a mean fluid flow present, also known as grazing flow. Propagating sound waves could lead to unsteady vortices shedding at the neck edge. These vorticities are then “convected away” with the mean flow. However, they act as a sound source near the upstream edge.

The resonant frequency of a given Helmholtz resonator was theoretically derived many years ago. However, the vorticity-involved noise-attenuating physics and mechanisms are not completely understood, especially when there is a grazing mean flow²³ (tangential “hot” or “cold” flow over the resonator neck). Some experimental attempts were made¹⁴ to visualize and to measure the noise damping mechanism and the performance of a Helmholtz resonator, as a hot grazing mean flow is present. Both linear and nonlinear acoustic responses of the resonator were measured. Experimental measurements of a Helmholtz resonator response were conducted by Ghanadi et al.²³ in a subsonic wind tunnel, which is used to produce a grazing cold turbulent flow. The resonator geometry was found to play a critical role on determining the excitation levels of acoustic velocity and pressure. Numerical investigations were also performed.²⁴ For example, Dai²⁴ studied the excitation of a Helmholtz resonator by convected vorticities.

Helmholtz resonators can be designed into symmetric and asymmetric configurations. Selamet and his group members^13,25–27 conducted a series of studies on circular resonators (either symmetric or asymmetric) via experimental measurements or theoretical/numerical simulations. It was worth noting that most of these studies, even when an asymmetric resonator was considered, were conducted, when there is no grazing mean flow. To increase the resonator’s damping effect, absorbing materials or perforated plates were inserted into the resonator neck. The overall damping performance was compared with those without insertions. In addition, modified designs of a Helmholtz resonator were proposed by extending its neck into its cavity. The extended neck was theoretically and numerically shown to affect both the TL and the resonance frequency. To validate this finding, an impedance tube was applied to conduct experimental measurements. The numerical and theoretical investigations were conducted by applying boundary element method and piston-driven model. However, these numerical and theoretical studies are quite challenging to capture vorticity-involved damping mechanisms and difficult to examine the grazing flow role. This motivated partially the present study.

Foregoing researches and studies show that the acoustic noise-attenuating performance and vorticity-involved physics of symmetric or asymmetric Helmholtz resonators were studied by time- or frequency-domain modeling via solving nonlinear Navier–Stokes equations²⁸ or performing laser-involved flow visualization experiments. Such experimental and numerical investigations are time consuming and costly. To decrease the computational time and cost, and to determine the noise-attenuating performance and vorticity-involved physics of an asymmetric Helmholtz resonator with a grazing flow present, computationally efficient numerical models are necessary. To the maximum knowledge of the present authors, there are few experimental and numerical studies discussed in the literature. This motivated partially the present study.

In this work, a 2D numerical model is developed by considering the linearized Navier–Stokes equations. The model is built in a frequency domain and it can be used to evaluate the noise-attenuating performances of asymmetric Helmholtz resonators with a grazing flow present. The conventional symmetric resonator design is chosen to be a benchmark/reference configuration. The asymmetric configuration is associated with an offset neck. The developed 2D model is first validated by comparing with the experimental data available in the literature. Further validation is conducted by comparing with 3D analytical results of an asymmetric resonator available in the literature. The present paper is organized as follows. In the following section, the 2D frequency-domain model with governing linearized Navier–Stokes equations is built. In “Results and discussion” section, the calculated results from the present model are presented and compared with the analytical and experimental ones. The effects of (1) the grazing flow Mach number and (2) the asymmetry on the noise-attenuating performances are evaluated over the frequency range of 50–200 Hz. Finally, the key findings on the asymmetric Helmholtz resonator are discussed and summarized in the “Conclusions” section.

Description of the numerical model

The current work is concerned with a cylindrical duct with an asymmetric Helmholtz resonator implemented as schematically illustrated in Figure 1. The asymmetry is characterized by the neck offset distance $e_{c} / a_{s}$ .

Figure 1.

Schematic drawing of the modeled cylindrical duct with (a) a conventional symmetric and (b) an asymmetric Helmholtz resonator attached.

The 2D model is built by solving the linearized Navier–Stokes equations for incompressible and viscous air flow at low Mach number. The governing equations include mass, momentum, and energy conservation ones in a Cartesian coordinate system as given as

\nabla \cdot (ρ_{0} u + ρ u_{0}) + \frac{\partial ρ}{\partial t} = S_{l s}

(1)

where

S_{l s}

denotes the acoustic fluctuations produced from a monopole source, such as a loudspeaker. The acoustic fluctuations are in superimposed sinusoidal waveforms

S_{l s} = \sum_{m = 1}^{M} λ_{m} \sin (ω_{m} t)

. The fluctuations are represented by velocity, pressure, density, and temperature (u, p, ρ, and

T

). Subscript 0 denotes the steady-state air flow parameters.

The momentum conservation holds as

\frac{\partial u}{\partial t} + (u_{0} \cdot \nabla) u + ρ (u_{0} \cdot \nabla) \frac{u_{0}}{ρ_{0}} = \frac{1}{ρ_{0}} \nabla \cdot σ - \frac{u_{0}}{ρ_{0}} S_{l s}

(2)

The energy conservation holds as

\begin{matrix} \frac{\partial T}{\partial t} + u \cdot \nabla T_{0} + u_{0} \cdot \nabla T + \frac{ρ}{ρ_{0}} (u_{0} \cdot \nabla T_{0}) - (\frac{α_{p} T}{ρ_{0} c_{p}}) u_{0} \cdot \nabla p_{0} \\ - \frac{α_{p} T_{0}}{ρ_{0} c_{p}} [\frac{\partial p}{\partial t} + u \cdot \nabla p_{0} + u_{0} \cdot \nabla p] = \frac{Φ + \nabla \cdot (κ \nabla T)}{ρ_{0} c_{p}} \end{matrix}

(3)

where

Φ

is the dissipation function, i.e.

Φ = τ_{i j} \partial u_{i} / \partial x_{j}

τ_{i j} = μ (\frac{\partial u_{i}}{\partial x_{j}} + \frac{\partial u_{j}}{\partial x_{i}} - \frac{2}{3} \frac{\partial u_{k}}{\partial x_{k}}) δ_{i j}

. The state equation is

p = ρ R T

. The stress tensor is given as

σ = - p I + τ = - p I + μ [\nabla u + {(\nabla u)}^{T}] + (μ_{B} - \frac{2}{3} μ) (\nabla \cdot u) I

. Here

ρ = ρ_{0} (β_{T} p - α_{p} T)

. τ is the viscous stress tensor, μ is the dynamic viscosity, and μ_B is the bulk viscosity. I is the identity matrix. The time derivative

\partial X / \partial t

of a given parameter

X

could be replaced with

j ω X

in frequency domain, i.e. a multiplication of

X

with

j ω

Turbulent motion leads to Reynolds stress, which is related to the fluctuating velocities’ cross-product terms. Thus, there is a need to model the turbulence. In the literature, there are various turbulence models such as Reynolds stress, eddy viscosity, large eddy simulation models to select. For simplicity, the current study chooses $k - ϵ$ turbulence model, since the interested Reynolds number is lower. k and ε denote turbulent kinetic energy and dissipation, respectively. They are determined by solving the transport equations, which are similar in structure to the governing equations for the air flow. The $k - ϵ$ turbulence models are described in the following two equations as

\frac{\partial (ρ k)}{\partial t} + \frac{\partial (ρ k u_{i})}{\partial x_{i}} = \frac{\partial}{\partial x_{j}} ((μ + \frac{μ_{t}}{σ_{k}}) \frac{\partial k}{\partial x_{j}}) + μ_{t} G_{k} - ρ ε - \frac{2}{3} (μ_{t} \frac{\partial u_{i}}{\partial x_{i}} + ρ k) + μ_{t} G_{b}

(4)

where $G_{k} = (\frac{\partial u_{i}}{\partial x_{j}} + \frac{\partial u_{j}}{\partial x_{i}}) \frac{\partial u_{i}}{\partial x_{j}}$ and $G_{b} = - \frac{ρ}{μ_{t}} \bar{u_{i}^{'} u_{j}^{'}} \frac{\partial u_{i}}{\partial x_{j}} - [G_{k} - \frac{2}{3} (\frac{\partial u_{i}}{\partial x_{i}} + \frac{k ρ}{μ_{t}})] \frac{\partial u_{i}}{\partial x_{i}}$ . $G_{b}$ and $G_{k}$ represent the production of turbulence kinetic energy due to buoyancy and mean velocity gradients

\begin{array}{l} \frac{\partial (ρ ε)}{\partial t} + \frac{\partial (ρ ε u_{i})}{\partial x_{i}} = \frac{\partial}{\partial x_{j}} ([μ + \frac{μ_{t}}{σ_{ε}}] \frac{\partial ε}{\partial x_{j}}) \\ + C_{1 ε} \frac{ε}{k} (μ_{t} [G_{k} + C_{3 ε} G_{b}] - \frac{2}{3} [μ_{t} \frac{\partial u_{i}}{\partial x_{i}} + k ρ] \frac{\partial u_{i}}{\partial x_{i}}) - ρ C_{2 ε} \frac{ε^{2}}{k} + R_{ε} \end{array}

(5)

$R_{ε} = ρ C_{3 ε} ε \frac{\partial u_{i}}{\partial x_{i}}$ . The model constants are $C_{1 ε} = 1.42$ , $C_{2 ε} = 1.68$ . α_k and $α_{ε}$ are inverse effective Prandtl number for k and ε, respectively.

Dimensions, geometry, and mesh

The geometry and dimensions of the modeled cylindrical duct in the presence of the mean flow and with the asymmetric Helmholtz resonator attached are summarized in Table 1.

Table 1.

Geometric dimensions of the cylindrical duct with a mean flow present and an asymmetric Helmholtz resonator implemented.

Parameters	Values	Parameters	Values	Parameters	Values
$L_{d}$	0.65 m	$V_{s}$	$4.5 \times 10^{- 3} m^{3}$	p ₀	$1.01325 \times 10^{5}$ Pa
$L_{u}$	0.65 m	$M_{u}$	$0 \to 0.10$	ρ ₀	$1.2 {kg/m}^{3}$
$D_{d}$	0.0486 m	$L_{s} = 2 a_{s}$	16.51 cm	$T_{0}$	297 K
$L_{n}$	0.0805 m	$e_{c} / a_{s}$	$0 \to 0.5$	λ_m	0.01 m/s
D_n	0.0404 m	$ω / 2 π$	$50 \to 200$ Hz	R	8.314

The mesh of the computational duct model is shown in Figure 2. To obtain an appropriate resolution of the vortices shedded and the shear layer at the neck–duct connected regions, a fine mesh with a total of 803,482 cells is chosen by conducting mesh independent study. The selected mesh is justified by validating the predicted results from the present model in comparison with the experimental and analytical results available in the literature, as discussed in the following section. By following initial trials, the time step is set to $1.0 \times 10^{- 4}$ s. It is applicable to all computational cases in the following sections, since it produces numerical results with an acceptable accuracy. The time step could be reduced further. However, this will lead to a dramatic increase of computational cost and time. However, decreasing the time step has little effect on TL performance over the interested frequency range.

Figure 2.

Computational unstructured mesh of the 2D model near the asymmetric resonator attached regions. (a) 634,956 cells and (b) 803,482 cells.

As plane waves are travelling along the duct with the asymmetric resonator implemented,both boundary conditions of the duct inlet and outlet are assumed to satisfy perfectly matched layer. The $TL$ of the duct with the resonator attached is determined as

TL (ω) \equiv 10 \log_{10} ({| | \frac{P_{i} (ω)}{P_{t} (ω)} | |}^{2})

(6)

where $P_{i} (ω)$ and $P_{t} (ω)$ denote the incident and transmitted acoustic waves in frequency domain on the upstream and downstream sections, respectively. When experimental measurements are conducted, the TL as defined above can be determined by applying the classical two-microphone technique.^29,30

Model validation

The linearized NS equations with the $k - ϵ$ turbulence model were solved by using COMSOL MULTIPHYSICS V5.3 with the UMFPACK direct solver. It is based on finite element methods. To evaluate the proposed model, validation studies are performed on a straight 2D duct with a symmetric Helmholtz resonator attached and the presence of a grazing flow. The results are then compared with the experimental measurements presented in the previous work by Selamet et al.²⁷ The parameters and values in the computational cases correspond to those of the tested cases and the geometry in the experimental measurement. The interested frequency range is from 50 to 200 Hz. This is consistent with the experimental measurement.²⁷ Figure 3 shows the comparison of the present predictions and the results obtained by Selamet et al.²⁷ under three different Mach numbers. It can be seen that good agreements are obtained in terms of the TL variation with forcing frequency.

Figure 3.

Validation of the numerical model by comparing the predicted results with the experimental ones available in Selamet et al.,²⁷ as M_u is set to three different values and $e_{c} / a_{s} = 0$ . (a) M_u = 0, (b) M_u = 0.05, and (c) M_u = 0.1. TL: transmission loss.

Further validation is conducted by considering the asymmetric Helmholtz resonator studied by Selamet and Ji.²⁶ Figure 4 shows the comparison of the present predictions and the 3D theoretical results obtained by Selamet and Ji,²⁶ as there is no grazing flow but the neck offset distance $e_{c} / a_{s}$ is set to five different values. It can be seen that the predicted resonant frequencies remain unchanged from the present model, as $e_{c} / a_{s}$ is varied. This is most likely due to the absence of the grazing flow. However, the 3D theoretical analysis predicted that the resonant frequency was decreased with increased $e_{c} / a_{s}$ by 1 Hz with respect to that corresponding to $e_{c} / a_{s} = 0$ . Correspondingly, there is an approximately 1/91.8 $\approx$ 1.09% variation of the resonant frequency. Such negligible frequency variation is not captured by the present model. However, the present predicted frequencies are in reasonably good agreement with the theoretical ones,²⁶ since the difference between the predicted and the theoretical results is less than 2 Hz for each given $e_{c} / a_{s}$ . This confirms that the present model is able to predict the noise damping performance of the asymmetric Helmholtz resonators, even in the presence of a nonnegligible grazing flow.

Figure 4.

Validation of the numerical model by comparing the predicted results with the 3D theoretical ones available in the Selamet and Ji,²⁶ as $e_{c} / a_{s}$ is set to five different values and M_u = 0.

Results and discussion

The numerical model is revised further to study the noise damping effect of asymmetric Helmholtz resonators in the presence of a nonnegligible grazing flow. The effect of the grazing flow is evaluated and shown in Figure 5. It can be seen that the maximum TL³¹ is decreased from 39 to 11 dB by approximately 28 dB, as the grazing flow Mach number is increased from 0 to 0.10, as the neck offset distance is set to five different values. The resonant frequency corresponding to the maximum TL is found to shift by approximately 10% to higher values with increased M_u but less than 0.07. Further increasing M_u is shown to give rise to the resonant frequency being reduced. Table 2 summarizes the resonant frequencies of the asymmetric resonator corresponding to the maximum TL.

Figure 5.

Variation of the TL with the forcing frequency, as the grazing flow Mach number is set to five different values. (a) $e_{c} / a_{s} = 0$ , (b) $e_{c} / a_{s} = 0.1$ , (c) $e_{c} / a_{s} = 0.2$ , (d) $e_{c} / a_{s} = 0.3$ , (e) $e_{c} / a_{s} = 0.4$ , and (f) $e_{c} / a_{s} = 0.5$ . TL: transmission loss.

Table 2.

A summary of the resonant frequency of the asymmetric Helmholtz resonator corresponding to the maximum transmission loss, as $e_{c} / a_{s}$ is set to five different values.

$e_{c} / a_{s} \ M_{u}$	$M_{u} = 0.0$	$M_{u} = 0.01$	$M_{u} = 0.05$	$M_{u} = 0.07$	$M_{u} = 0.10$
0	90.0	90.0	100.0	105.0	105.0
0.1	90.0	90.0	100.0	105.0	105.0
0.2	90.0	90.0	100.0	105.0	100.0
0.3	90.0	90.0	100.0	105.0	100.0
0.4	90.0	90.0	100.0	105.0	100.0
0.5	90.0	90.0	100.0	105.0	95.0

The resonant frequency Ω _r of the asymmetric Helmholtz resonator could also be predicted by using the classical formula as

\frac{Ω_{r}}{2 π} = \frac{c_{0}}{2 π} \sqrt{\frac{π D_{n}^{2}}{4 V_{s} (L_{n} + L_{c})}}

(7)

where

V_{s}

is the cavity volume.

L_{n}

and D_n denote the neck length and diameter, respectively.

L_{e f f} \equiv L_{n} + L_{c}

is known as the effective neck length.

L_{c}

is the length correction, which characterizes the inertia effect. Note that Ω _r predicted from equation (7) is approximately 91.0 Hz, which agrees well with the numerical predicted results at M_u = 0. However, as M_u and

e_{c} / a_{s}

are increased, the above equation fails in predicting the resonant frequency, since equation (7) does not consider the effects of the mean grazing flow and the geometric shape. The empirical prediction from equation (7) is found to lead to approximately 16% difference from the present numerical results.

To shed lights on the TL and to gain insights on the vortex shedding and shear layer separation near the neck surface, Figure 6 illustrates the 2D contours of vorticity and velocity, as $e_{c} / a_{s}$ is set to three different values. It can be seen that the neck offset distance $e_{c} / a_{s}$ dramatically affects the vortex shedding intensity and its distributions. The shear separation is also significantly different.³² This may be due to the fact that nonslip boundary conditions³³ hold on the neck surface.

Figure 6.

Comparison of the vorticity (a)–(c) and velocity (d)–(f) contours near the offset neck, as $M_{u} = 0.1$ and $ω / 2 π = 100$ Hz. (a) and (d) $e_{c} / a_{s} = 0$ , (b) and (e) $e_{c} / a_{s} = 0.2$ , (c) and (f) $e_{c} / a_{s} = 0.5$ .

The effect of asymmetry, i.e. the neck offset distance $e_{c} / a_{s}$ on the noise-attenuating performance is examined. This is shown in Figure 7. It can be seen that $e_{c} / a_{s}$ leads to the resonant frequency³⁴ being decreased and so the maximum TL, as a grazing flow²³ is present.

Figure 7.

Variation of the TL of the asymmetric resonator with the forcing frequency, as the neck offset distance $e_{c} / a_{s}$ is set to six different values. (a) M_u = 0, (b) $M_{u} = 0.01$ , (c) $M_{u} = 0.05$ , (d) $M_{u} = 0.07$ , and (e) $M_{u} = 0.10$ .

The maximum $T L_{max}$ of the asymmetric resonators is illustrated in Figure 8. It can be seen that when M_u = 0, $T L_{max}$ is decreased with increased $e_{c} / a_{s}$ . However, when $M_{u} \geq 0.1$ , $T L_{max}$ is increased with increased $e_{c} / a_{s}$ . These findings confirm that both $e_{c} / a_{s}$ and M_u play important roles in determining $T L_{max}$ and so the combustor stability.^2,35,36

Figure 8.

Variation of the maximum TLs of the asymmetric resonator with the Mach number M_u and $e_{c} / a_{s}$ . TL: transmission loss.

Conclusions

In the present study, 2D numerical model of a cylindrical duct with an asymmetric Helmholtz resonator attached is developed in frequency domain and applied to predict its TL performance. The asymmetry effect is examined by considering five different neck offset distances from the center of the resonator cavity denoted by $e_{c} / a_{s}$ , as a grazing flow is present. Both acoustics and vorticity-involved flow behaviors are successfully captured. The model is first validated with comparison with the experimental measurements obtained from a flow impedance tube available in the literature. Further validation is conducted by comparing the present results with the analytical results of an asymmetric Helmholtz resonator available in the literature. After these validations, the frequency-domain model is further developed to study the noise damping performance of the asymmetric Helmholtz resonators in the presence of a variable Mach number of the grazing flow. The main observations from the present study are summarized as follows:

As the grazing flow Mach number M_u is small ( $M_{u} \leq$ 0.07), the resonant frequency of the asymmetric Helmholtz resonator is shown from the present simulations to be unrelated to the neck offset distance $e_{c} / a_{s}$ . However, as the Mach number is increased to $M_{u} = 0.1$ , it is found that increasing the neck offset distance $e_{c} / a_{s}$ leads to a decrease of the resonant frequency corresponding to the maximum TL. This variation may be due to the nonnegligible grazing flow effect. Further experimental studies are needed to confirm these findings and the role of $e_{c} / a_{s}$ on resonant frequency.

Increasing the grazing flow Mach number leads to the maximum TL being reduced by more than 60% (see Figure 7(a) and (e)). The mean grazing flow, in general, gives rise to a weaker interaction between the asymmetric resonator cavity and the main pipe. Thus, it results in the resonant frequencies to higher values but a reduced maximum TL.

The classical formula $Ω_{r} = c_{0} \sqrt{\frac{π D_{n}^{2}}{4 V_{s} (L_{n} + L_{c})}}$ fails in predicting the resonant frequencies of the asymmetric Helmholtz resonators, especially at a higher grazing flow Mach number.

Intensity of vortex shedding is increased near the edge of the offset neck, as M_u is increased. This reveals the increased shear separation on the surface of the neck. This leads to the diminished capacity of the resonator for attenuating unwanted noise at higher grazing flow velocities.

In summary, the asymmetric Helmholtz resonator is numerically studied, with the overall aim of increasing its noise-attenuating performance to enable it being implemented in engine combustors.^37–42

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/ or publication of this article: This work is supported by the National Natural Science Foundation of China with grant nos. 51506079 and 11661141020. This financial support is gratefully acknowledged.

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Numerical studies of transmission loss performances of asymmetric Helmholtz resonators in the presence of a grazing flow

Abstract

Keywords

Introduction

Description of the numerical model

Dimensions, geometry, and mesh

Model validation

Results and discussion

Conclusions

Footnotes

Declaration of conflicting interests

Funding

References