A mathematical formulation for the optimal impedance of a curved axial microperforated panel in a duct bend

Yang, Cheng

doi:10.1121/10.0005759

Previous work [J. Acoust. Soc. Am. 145, 501–511 (2019)] showed that a micro-perforated panel (MPP) placed in a duct bend will couple to the separated air channels to constitute a resonant system. This coupling effect results in the vibration of air mass inside micropores and provides damping to the duct. However, only limited attenuation was obtained. To enhance the damping effect, a mathematical formulation for the optimal impedance of the MPP (or a thin structure that has continuous normal velocity and pressure jump boundary conditions) is derived based on the concept of Cremer impedance which is mathematically equivalent to finding the exceptional point of a non-Hermitian system. Validation of the proposed model is made by examining the mode merging phenomenon at the exceptional point. The optimal damping is also compared with a straight duct having one of its wall lined.

1. Introduction

Designing compact acoustic treatment for duct noise reduction has always been a challenging task. A common practice is lining duct walls with absorption materials. Recently, the concept of modal filter which features a number of MPPs placed axially and radially inside a duct was proposed.^2–5 The underlying mechanism is producing damping by placing MPPs at position where large acoustic pressure gradient of a given duct mode occurs. The pressure difference between two sides of the MPP causes a strong air oscillation inside micropores and damps duct modes. For a straight duct, the modal filter has been proved effective for damping high-order duct modes. Yang^1,6 extended the application of modal filter to a duct bend in the low frequency range for the no flow case. However, experiment showed limited attenuation⁶ due to the moderate pressure variation over the duct cross section. In order to enhance the attenuation, the concept of Cremer impedance^7–9 is applied to the modal filter to find the optimal impedance that maximizes the modal damping. The Cremer impedance refers to an impedance value that results in a double root of the eigen equation. This is essentially the same as the exceptional point of a non-Hermitian system in quantum physics.¹⁰ The idea has recently drawn the interest of physical acoustics community for acoustic metasurface design.¹¹

2. Mathematical derivation

2.1 Eigenvalue equation

Figure 1 shows a two-dimensional duct bend with its outer and inner radii being R₁ and R₂, respectively. To avoid the scattering that may arise due to impedance discontinuity, the bend is assumed to extend circumferentially into a complete angular region. A curved MPP (dashed line) at r = R₁₂ divides the bend into domains 1 and 2, and the acoustic pressure of each is

p_{1} = [M_{1} J_{β} (k r) + N_{1} Y_{β} (k r)] e^{- j β θ}

(1)

and

p_{2} = [M_{2} J_{β} (k r) + N_{2} Y_{β} (k r)] e^{- j β θ},

(2)

where $J_{β} (k r)$ and $Y_{β} (k r)$ are the Bessel functions of the first kind and second kind, k the free-field wavenumber, and the order $β$ is the axial (or circumferential) wavenumber. The radial acoustic velocity vanishes at outer and inner walls (acoustically rigid), thus

M_{1} J_{β}^{'} (k R_{1}) + N_{1} Y_{β}^{'} (k R_{1}) = 0

(3)

and

M_{2} J_{β}^{'} (k R_{2}) + N_{2} Y_{β}^{'} (k R_{2}) = 0.

(4)

FIG. 1.

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Schematic of a duct bend containing a curved axial MPP.

The prime denotes derivative with respect to the argument of the Bessel function. Combining the above equations, one has

p_{1} = M_{1} [J_{β} (k r) - \frac{J_{β}^{'} (k R_{1})}{Y_{β}^{'} (k R_{1})} Y_{β} (k r)] e^{- j β θ} = M_{1} ϕ (β, k r) e^{- j β θ}

(5)

and

p_{2} = M_{2} [J_{β} (k r) - \frac{J_{β}^{'} (k R_{2})}{Y_{β}^{'} (k R_{2})} Y_{β} (k r)] e^{- j β θ} = M_{2} φ (β, k r) e^{- j β θ} .

(6)

For negligible thickness of the MPP in practice, the radial acoustic velocity at r = R₁₂ is assumed to be continuous across the panel thickness, yielding

M_{2} = \frac{J_{β}^{'} (k R_{12}) - \frac{J_{β}^{'} (k R_{1})}{Y_{β}^{'} (k R_{1})} Y_{β}^{'} (k R_{12})}{J_{β}^{'} (k R_{12}) - \frac{J_{β}^{'} (k R_{2})}{Y_{β}^{'} (k R_{2})} Y_{β}^{'} (k R_{12})} M_{1} = \frac{φ^{'} (β, k R_{12})}{ϕ^{'} (β, k R_{12})} M_{1} = T M_{1} .

(7)

In addition, the MPP yields a pressure jump boundary condition, which gives

\frac{p_{2} - p_{1}}{Z_{MPP}} = v_{r, 2} = v_{r, 1}, at r = R_{12},

(8)

where Z_MPP is the normal acoustic impedance of the MPP. Combining Eqs. (5)–(8), one has the eigenvalue equation for an axially infinite duct bend containing a curved MPP¹

T φ (β, k R_{12}) - ϕ (β, k R_{12}) = - \frac{Z_{MPP}}{j ω ρ} ϕ^{'} (β, k R_{12}) .

(9)

Given the MPP impedance and frequency, the transcendental equation can be solved numerically to obtain the axial wavenumber $β$ which is, in general, a complex value. The imaginary part corresponds to the decay rate of a mode in the propagating direction.

2.2 The optimal impedance

Following Eq. (9), one defines a function f as

f (β, k) = \frac{ϕ^{'} (β, k R_{12})}{T φ (β, k R_{12}) - ϕ (β, k R_{12})} + j k A_{MPP}^{'},

(10)

where $A_{MPP}^{'} = ρ c / Z_{MPP}$ is the dimensionless acoustic admittance of the MPP. For a straight duct, the optimal impedance occurs when the transverse wavenumbers of two neighboring modes merge in the complex plane and the eigenvalue represents the double root. For the duct bend in current work, the problem becomes finding the axial wavenumber that makes the first order derivative of $f (β, k)$ with respect to $β$ vanish at a given k, yielding

\frac{(A - B) (C - D) [(\frac{\partial G}{\partial β} - \frac{\partial H}{\partial β}) (E - F) + (\frac{\partial E}{\partial β} - \frac{\partial F}{\partial β}) (G - H)]}{{(E - F)}^{2} {(G - H)}^{2}} = 0,

(11)

where

\begin{matrix} A = J_{β}^{'} (k R_{12}) Y_{β}^{'} (k R_{1}), \\ B = J_{β}^{'} (k R_{1}) Y_{β}^{'} (k R_{12}), \\ C = J_{β}^{'} (k R_{2}) Y_{β}^{'} (k R_{12}), \\ D = J_{β}^{'} (k R_{12}) Y_{β}^{'} (k R_{2}), \\ E = J_{β}^{'} (k R_{2}) Y_{β}^{'} (k R_{1}), \\ F = J_{β}^{'} (k R_{1}) Y_{β}^{'} (k R_{2}), \\ G = J_{β} (k R_{12}) Y_{β}^{'} (k R_{12}), \\ H = J_{β}^{'} (k R_{12}) Y_{β} (k R_{12}) . \end{matrix}

To find roots of Eq. (11), one needs to calculate the Bessel functions with a complex order. The Bessel function of the first kind with a complex order $β$ is related to the confluent hypergeometric function as^12,13

J_{β} (k r) = \frac{{(k r / 2)}^{β} e^{- jkr} Ψ (β + 1 / 2; 2 β + 1; 2 j β)}{Γ (1 + β)}

(12)

and for the Bessel function of the second kind, it is

Y_{β} (k r) = - π^{- 1 / 2} {(2 k r)}^{β} [e^{j (β π - k r)} Ψ (1 / 2 + β; 1 + 2 β; 2 j k r) + e^{j (k r - β π)} Ψ (1 / 2 + β; 1 + 2 β; - 2 j k r)],

(13)

where $Γ (\cdot)$ is the Gamma function and $Ψ (\cdot; \cdot; \cdot)$ is the confluent hypergeometric function. In addition, finding solutions to Eq. (11) also requires the calculation of the derivative of the Bessel function with respect to its order. The closed expressions for the first derivative of $J_{β} (k r)$ and $Y_{β} (k r)$ with respect to any value of $β$ are given by Brychkov¹⁴ as

\begin{matrix} \frac{\partial J_{β} (k r)}{\partial β} = \frac{π}{2 Γ^{2} (β + 1)} {(\frac{k r}{2})}^{2 β} [Y_{β} (k r) - cot (β π) J_{β} (k r)] \\ {}_{2}Ψ_{3} (\begin{matrix} β, β + 1 / 2; - {(k r)}^{2} \\ β + 1, β + 1, 2 β + 1 \end{matrix}) + J_{β} (k r) [\frac{1}{2 β} - ψ (β + 1) + ln \frac{k r}{2} + \frac{{(k r)}^{2}}{4 (β^{2} - 1)} {}_{3}Ψ_{4} (\begin{matrix} 1, 1, 3 / 2; - {(k r)}^{2} \\ 2, 2, 2 - β, β + 2 \end{matrix})] \end{matrix}

(14)

and

\begin{matrix} \frac{\partial Y_{β} (k r)}{\partial β} = - π cot (β π) Y_{β} (k r) + f_{- β} (k r) - cos (β π) f_{β} (k r) \\ + J_{β} (k r) [cot (β π) g_{β} (k r) - π] + csc (β π) J_{- β} (k r) g_{- β} (k r) \\ + \frac{{(k r)}^{2}}{4 (β^{2} - 1)} [2 cot (β π) J_{β} (k r) - Y_{β} (k r)] {}_{3}Ψ_{4} (\begin{matrix} 1, 1, 3 / 2; - {(k r)}^{2} \\ 2, 2, 2 - β, 2 + β \end{matrix}), \end{matrix}

(15)

where

\begin{matrix} f_{β} (k r) = \frac{{(k r / 2)}^{2 β}}{2 π} Γ^{2} (- β) J_{- β} (k r) {}_{2}Ψ_{3} (\begin{matrix} β, β + 1 / 2; - {(k r)}^{2} \\ 2 β + 1, β + 1, β + 1 \end{matrix}), \\ g_{β} (k r) = ln (k r / 2) - ψ (β) - 1 / (2 β), \end{matrix}

and $ψ (\cdot)$ is the logarithmic derivative of the gamma function. The re-expression of the special functions (12)–(15) allows Eq. (11) to be solved by a standard root finding scheme. Once the axial wavenumber $β$ is obtained for a given frequency, it is substituted back into Eq. (10) for the optimal impedance.

3. Validation

Validation of the proposed formulation is made by examining the merging of two neighboring modes. First, one compute the optimal impedance Z_opt from Eqs. (10) and (11). Then, the axial wavenumbers of the first and second modes are computed iteratively by fixing the real part of Z_opt whilst varying its imaginary part from a sufficiently small value (–10) to a sufficiently large value (10). The axial wavenumber trajectories are depicted in Fig. 2 for two randomly selected frequencies. The blue cross-denotes the intersection between trajectories and a further check of the impedance shows this point corresponds to the condition Im(Z) = Im(Z_opt), confirming the occurrence of the mode merging at the optimal impedance.

FIG. 2.

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Axial wavenumber trajectories (red circles) of the first and second modes in the complex β plane. (a) 900 Hz, Z_opt = 1.9776 − 0.0899j, β_opt = 1.337 − 0.45j; (b) 1500 Hz, Z_opt = 1.3042 − 0.808j, β_opt = 2.443 − 0.717j. Blue cross: intersection of trajectories. Duct bend geometry: R₁ = 0.135 m, R₂ = 0.035 m, and R₁₂ = 0.085 m.

4. Comparison with a straight duct

The optimal impedance is compared with a two-dimensional straight duct with one wall lined. The latter is calculated based on the method in Ref. 7. A noticeable difference can be identified at low frequencies in Fig. 3. Small resistance and reactance (in absolute value) are required for maximum modal attenuation for the straight duct while large values are required for the duct bend. When the frequency increases, both the optimal resistance and the reactance (in absolute value) increase for the straight duct. But for the duct bend the optimal resistance and reactance constantly decrease. Note that a negative reactance (air mass) is required for the duct bend when kd > 1.6, which cannot be realized by an MPP alone.

FIG. 3.

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Optimal impedances of the straight and curved ducts as a function of the non-dimensional frequency kd, where d = R₁ − R₂ and d corresponds to the height for the straight duct case. Duct bend geometry: R₁ = 0.135 m, R₂ = 0.035 m, and R₁₂ = 0.085 m.

Comparison of the optimal axial wavenumber for the two ducts is depicted in Fig. 4. The real part of the optimal axial wavenumber shows a growing trend for both ducts. For the imaginary part, the straight duct shows a large modal decay rate at low frequencies which gradually decreases with the increase in frequency. However, the duct bend shows a much smaller modal decay rate at low frequencies. When the frequency increases, the modal decay rate increases until reaching a maximum at around kd = 4.6, corresponding to 2 dB transmission loss for an MPP with an axial length of 0.1 m (duct height). After that, the modal decay rate decreases moderately with the increase in frequency. From Fig. 4, it is inferred that, at low frequencies, the optimal damping of the duct bend containing a curved MPP is smaller than that for a straight duct having one lined wall while they are nearly identical at high frequencies.

FIG. 4.

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The real and imaginary parts of the optimal axial wavenumbers for the curved and straight ducts as a function of the dimensionless frequency. The right hand side y axis shows the transmission loss per duct height obtained from the imaginary part of the optimal axial wavenumber using $T L = 20 \log_{10} [\exp [- Im (β_{o p t} d)]]$ (for the damping material with an axial length of d).

5. Conclusions

The optimal impedance for a curved axial MPP contained in a duct bend is derived and validated. Compared with the straight duct, greater resistance and reactance are required for maximum damping at low frequencies. In view of the low axial decay rate for the duct bend case, tuning the impedance to the optimal impedance is of less importance. At high frequencies, the optimal damping achieved in both ducts are nearly identical, but a negative reactance (air mass) is required for the duct bend. The formulation presented can be extended to practical applications such as perforated vanes or lined bend.

Acknowledgment

This work was supported by the National Natural Science Foundation of China (Grant No. 12004247) and the State Key Laboratory of Mechanical System and Vibration (Grant No. MSVZD201903). The author is grateful to Professor Mats Åbom of KTH for many stimulating discussions.

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Author(s)

A mathematical formulation for the optimal impedance of a curved axial microperforated panel in a duct bend

1. Introduction

2. Mathematical derivation

2.1 Eigenvalue equation

2.2 The optimal impedance

3. Validation

4. Comparison with a straight duct

5. Conclusions

Acknowledgment

References and links

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A mathematical formulation for the optimal impedance of a curved axial microperforated panel in a duct bend

1. Introduction

2. Mathematical derivation

2.1 Eigenvalue equation

2.2 The optimal impedance

3. Validation

4. Comparison with a straight duct

5. Conclusions

Acknowledgment

References and links

Citing articles via

Submit your article

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Resources

Explore

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