Nonlinear effects on tonal sound in lined ducts

Abstract

Porous sound absorbers are a familiar and supposedly well-understood feature of the noise control palette. As in many other aspects of engineering, however, aerospace applications require a degree of mastery well beyond the common place: sound absorbers must be designed to be effective at high noise levels within strict mass and volume constraints. In the process of fine-tuning sound absorbers for jet aircraft, it was discovered that high incident sound pressure levels cause the performance of lined ducts to diverge from small-signal predictions. The underlying physics was not fully understood prior to the early 1980s, at which point NASA-sponsored research disclosed and quantified the primary mechanism. Further research by others continues building on that foundation down to the present. This paper gives a brief overview of the topic, summarizes one of the research efforts, and adapts its findings for lined duct applications.

Keywords

nonlinear acoustics sound absorption porous materials duct liners

Introduction

Porous materials regularly exhibit amplitude-dependent properties in high sound pressure level environments such as jet engine inlets.^1,2 Sound absorption occurs chiefly as a consequence of aerodynamic drag within the material. The drag manifests itself as flow resistance, which turns out to be strongly amplitude-dependent. The amplitude dependence and its consequences are the subject of this paper.

Kuntz¹ provides a detailed review of porous sound absorber research prior to 1982. Zwikker and Kosten’s model³ provides the basis for this analytical treatment. Lambert⁴ showed that, above a certain frequency inherent to the material, motions of the air and flexible absorber structure are decoupled. Kuhl and Meyer⁵ provided the key insight that the dc and ac flow resistance values are essentially interchangeable, and confirmed that the material frame can be considered rigid in many cases. Hersh and Walker^6,7 added empirical relations between viscosity, porosity, fiber diameter, and resistivity, along with heat transfer effects. Others such as Biot⁸ and Lambert⁹ developed comprehensive theories that describe absorber performance in the linear flow regime.

Zorumski and Parrot¹⁰ confirmed that ac and dc flow resistivities are functionally interchangeable, and that the resistivity values are essentially frequency independent. They also observed the increasing impedance of the material at high amplitudes, as well as unexplained third-harmonic distortion products radiating from the air–absorber interface. These nonlinear effects appeared at levels far below those commonly associated with nonlinearity in air, and called into question design methods based on the linear theory.

Kuntz’s research¹ accurately predicted nonlinear effects including amplitude-dependent acoustic impedance and amplitude saturation. Other observed behaviors continued to defy explanation. In particular, pure tones propagating within a porous material developed third harmonics that increased with distance before decaying. In fact, the third harmonic level exceeded that of the second at many points. Airborne nonlinear acoustics models suggest that shock steepening would cause all harmonics to behave in this manner, but the second harmonic decayed monotonically like the fundamental. A different formulation was required.

Nelson¹¹ continued Kuntz’s research and approach, and appears to have been first to apply a Forchheimer-type nonlinearity¹² to the amplitude-dependent flow resistivity. Approximate solutions to the resulting nonlinear wave equation successfully predict observed wave propagation behaviors within the material, suggesting that the key nonlinear mechanism had been captured. An approximate analytical solution for initially pure tones was extended to permit evaluation of reflections from layers as well as the performance of lined ducts.

Later research in the 1980s and 1990s in the United States and France confirmed the Forchheimer nonlinearity and provided detailed formulations for amplitude-dependent complex flow resistance, density, and bulk modulus. The focus shifted to numerical and approximate analytical solutions for surface reflections.^13–17 In particular, the model of Wilson, McIntosh, and Lambert¹³ appears to have become the basis for much of the work that followed.

Beginning in the United Kingdom in the 2000s, a shift in emphasis occurred from the fibrous materials previously studied to granular materials and porous metals. The theory was extended to include complex compressibility and boundary slip,^18,19 multilayer and double-porosity materials,^20,21 and time-domain solutions and pulse propagation.^22,23 Recent work on porous metals continues in China.^24,25 The Forchheimer nonlinearity approach has also been applied to micro-perforate absorbers^26–29 as well as propagation over the ground.³⁰

In contrast to many other formulations, Kuntz’s streamlined approach suits itself well to the practical need of engineers by requiring a minimum number of parameters. This paper recapitulates the key points of this approach and extends their consequences to the performance of lined ducts.

Experimental results refer to samples of batted Kevlar® 29 with porosity of 0.96, flow resistivity of approximately 29,500 mks rayls/m, and nonlinear flow resistivity of approximately 6900 mks rayls s/m².

Propagation within the material

Simplifying assumptions are used to develop the equations of motion. Conservation of mass and momentum within the rigid-framed material are

\frac{\partial ρ}{\partial t} + ρ 0 \frac{\partial u}{\partial x} = 0

and

Ω Γ ρ 0 \frac{\partial u}{\partial t} + \frac{\partial p}{\partial x} + r (u) u = 0

in which Γ is the structure factor and Ω is the porosity. The dc flow resistivity r (with Forchheimer nonlinearity) is substituted for the ac flow resistivity

r (u) = r 1 + r 2 | u |

Complex compressibility due to heat transfer effects, especially in the presence of turbulent mixing at high sound levels, is assumed to push the state equation towards the isothermal limit of

c = c i = \frac{c 0}{\sqrt{γ}}

within the material.

Assuming the sound to consist of a tonal complex dominated by the fundamental, the wave equation for the nth harmonic in specific velocity V_n can be expressed as³¹

\frac{\partial 2 V n}{\partial χ 2} + q_{n}^{2} V n = jnR 2 \sum_{n = - \infty}^{\infty} V p W n - p

where χ is the nondimensional distance ωx/c_i, the specific wavenumber q_n is defined as

q n = q nr - jq ni and q n = n \sqrt{Ω Γ - j \frac{R 1}{n}}

the resistance terms R₁ and R₂ are

\frac{r 1}{ρ 0 ω}

and

\frac{r 2 c i}{ρ 0 ω}

, respectively, and W is the tonal spectrum of u sgn(u). For a sinusoid of amplitude A₁ and phase

φ v 1

, W is the spectrum of a square wave, which possesses only a DC term and even harmonics

W n = \frac{2 A 1}{π (1 - n 2)} \exp (- jn φ v 1 (χ))

For progressive waves of slowly varying amplitude, the fundamental acquires excess attenuation through V₁W₀ and V₋₁W₂ interaction, and can be shown to obey

\frac{\partial A 1}{\partial χ} + j (q 1 - \frac{\partial φ v 1}{\partial χ}) A 1 \approx - {Kq}_{1}^{*} A_{1}^{2}

where the nonlinearity coefficient K is

K = \frac{4}{3 π} \frac{R 2}{| q 1 |}

The real and imaginary parts of this equation lead respectively to solutions for the fundamental tone amplitude

A 1 (χ) = \frac{A 1 (0) \exp (- q 1 i χ)}{1 + \frac{KA 1 (0)}{Q 1} (1 - \exp (- q 1 i χ))}

and phase velocity

C v 1 = (\frac{\partial φ v 1}{\partial χ}) - 1 \approx \frac{1}{q 1 r + Kq 1 i A 1}

Excess attenuation at high amplitudes arises from increased flow resistance as well as energy shifted into higher, odd harmonics. Amplitude saturation occurs at high amplitudes when excess attenuation balances increased amplitude, as documented in air by Thuras et al.³² and in porous materials by Kuntz.¹ The collection of parameters KA₁₀/Q₁ corresponds to the Gol'dberg number Γ in the familiar solution to Burgers’ equation.³³

The phase velocity, already slow at low frequencies due to the effect of diffusion, is further reduced with amplitude.

Figure 1 depicts results of propagation measurements (after Figures 4 to 11 of Zorumski and Parrot¹⁰) showing excess attenuation of the 1^st (and 2^nd) harmonics of an initially pure tone. The initial growth of the 3^rd harmonic will be addressed later.

Figure 1.

Propagation of tonal harmonics.

Reflection from the absorber surface

The specific impedance experienced by the wave consisting of harmonics with amplitude A_n and phase $φ vn$ is derived from the continuity equation as

Z n = \frac{1}{n} (\frac{\partial φ vn}{\partial χ} + j \frac{1}{A n} \frac{\partial A n}{\partial χ})

so that for a progressive tone with slowly varying amplitude

Z 1 = q 1 - {jKq}_{1}^{*} A 1

Figure 2 shows the amplitude dependence of impedance and phase velocity.

Figure 2.

(a) Specific impedance vs. amplitude; (b) specific phase velocity (c_ph/c_i) vs. amplitude.

We assume that the slowly varying progressive wave solution is adequate, overlooking the complex interaction of counter-propagating waves in the porous layer. On that basis, the oblique incidence surface impedance can be shown to be

ξ = - j \frac{Z}{Ω \sqrt{γ}} \frac{q \cos θ}{q χ (θ)} \cot (q χ (θ) d)

where

q χ (θ) = \sqrt{q 2 - \sin 2 θ}

and all values refer to the fundamental (subscript n = 1). It should be noted that the amplitude dependence manifests itself in the impedance term (Figure 3).

Figure 3.

(a) Amplitude-dependent normal incidence sound absorption vs. frequency and (b) vs. flow resistivity.

Figure 4.

(a) Dimensionless design chart IL/h vs. 2hf/c₀, and (b) IL vs. frequency.

Sound absorption of the layer is determined in the conventional manner from

R = \frac{ξ - 1}{ξ + 1}

and

α = 1 - | R | 2

There exists an ideal flow resistivity for a porous layer of a particular thickness, which maximizes sound absorption at a given frequency. For high frequencies and/or adequately thick absorbers, the additional impedance due to high amplitude is detrimental and reduces absorber performance. However, for thin layers and/or low frequencies, the additional impedance may lead to increased sound attenuation.

Lined ducts

The performance of a lined duct is estimated by finding approximate solutions to the transcendental equation³⁴

k χ h \tan (k χ h) = \frac{Ω γ}{qZ} \frac{h}{d} [q χ d \tan (q χ d)]

Mechel³⁵ has developed approximate solutions for the least-attenuated mode given a locally reacting liner. The equations are cast in terms of the quantity U, which is the product of wavenumber half-depth of the duct (k₀h) and the specific admittance of the absorber surface (Z₀G). Adapting our notation, the quantity U can be shown to be

U = - j \frac{Ω}{Z} \sqrt{γ} k 0 h \tan (qd)

Approximate solutions for the least attenuated mode in a square duct lined on four sides, due to the nonzero real part of the downstream wavenumber Γ, are

z circ = \frac{105 + 45 jU \pm \sqrt{11025 + 5250 jU - 1505 U 2}}{20 + 2 jU}

and

Γ ij = \sqrt{2 ɛ ij - k 2}

where

ɛ ij = \frac{\sqrt{z ij}}{h}

Four wavenumbers are possible for each value of z. Care must be taken to select the branch with the smallest attenuation per distance.

Figure 4 shows the amplitude-dependent insertion loss IL of a square duct having 51 mm half-width h and a 51 mm liner having the same properties as the tested samples. The overall length of the lined duct is 305 mm.

Drastic performance degradation is apparent at high amplitudes. These predictions coincide with the early observations of sound absorbers under-performing their design expectations. The overall attenuation performance drops, while the tuning broadens and moves towards higher frequency. The appropriate design response is to compensate by preferring less dense materials.

As an exercise, we estimate the performance of the same arrangement with half the flow resistivity, as depicted in Figure 5. The reduced linear flow resistivity (r₁ = 15000 mks rayls/m) is closer to the optimal value for this thickness and a 1000 Hz tone (see Figure 3(b)), so the overall attenuation improves. Even taking into account nonlinearity at amplitudes as high as 157 dB (r₂ = 3500 mks rayls s/m²), the less-dense material outperforms the original, small-signal results above 1 kHz.

Figure 5.

(a) Dimensionless design chart for one-half resistance, and (b) IL vs. frequency for reduced resistance.

Harmonic distortion

Harmonic distortion is another consequence of nonlinear flow resistivity. In fact, odd harmonic distortion products appear as the wave progresses, which grow in strength before decaying. The third harmonic is strongest and is driven by V₁W₂ and V₋₁W₄ interaction within the material, which then radiates from the interface back into the lined duct.¹⁰

The lined duct muffler is usually tuned to suppress the primary; other frequencies are attenuated less rapidly along the duct. Thus, harmonic distortion must be considered as well to avoid compromising muffler performance.

For a strong fundamental of slowly varying amplitude, the third harmonic within the material obeys

\frac{\partial A 3}{\partial χ} + j (q 3 - \frac{\partial φ v 3}{\partial χ}) A 3 \approx \frac{4 R 2}{5 π q 3} A_{1}^{2}

The real part, which addresses tonal amplitude, reduces to

\frac{\partial A 3}{\partial χ} + q 3 i A 3 \approx \frac{4 R 2}{5 π | q 3 |} q 3 r A_{1}^{2}

Although the third harmonic amplitude A₃ may be small when the tone strikes the interface, its gradient (from which the pressure amplitude is derived) need not be. Thus the third harmonic specific pressure Π₃ is

Π 3 \approx 2 \frac{\partial A 3}{\partial χ} | x = 0 \approx \frac{2}{5} \frac{| q 1 |}{| q 3 |} Kq 3 r A_{10}^{2}

Table 1 gives computed sound pressure levels for a third harmonic generated at the interface due to various incident fundamental tone levels. The increasing strength of the harmonic at high amplitudes corroborates experimental observations by Zorumski and Parrot.²

Table 1.

Estimated third-harmonic levels due to nonlinear interaction at interface.

Incident level (dB)	Reflected harmonic level (dB)
118	61
138	101
159	138
169	153
180	167

The tabulated values roughly follow the equation

L p, 3 \approx - 283 + 3.69 L p, 1 - 0.00656 L_{p, 1}^{2}

Conclusion

Amplitude-dependent flow resistance affects impedance and, consequently, sound absorption and duct attenuation. These effects become noticeable above 140 dB for the materials studied. Generally, the increase in impedance causes a decrease in absorber performance. However, at lower frequencies or for thin absorber layers, the additional amplitude-induced impedance may cause the performance to improve somewhat.

Approximate solutions for progressive tones with slowly varying amplitude have been combined with lined duct estimation methods to allow the study of lined duct performance at high sound pressure levels.

Footnotes

Acknowledgement

It’s difficult to know how one’s work will affect the lives of others. Edward J. Rice was contract monitor for NASA Science Grant 3198 beginning in 1978. Two of the theses that it funded (Kuntz, and Nelson) advanced the understanding of nonlinear porous absorbers and provided a foundation for work that continues to this day. Herbert L Kuntz, II blazed a trail of meticulous analysis and experimental design and execution that the author was fortunate to inherit. Whatever success occurred in the author’s research is largely due to Kuntz’s efforts. David T Blackstock, mentor and thesis advisor, helped edit the author’s thesis with remarkable patience, and provided excellent guidance to his students and colleagues by virtue of his keen insight and encyclopedic knowledge of acoustics.

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) received no financial support for the research, authorship, and/or publication of this article.

Appendix

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