Vibroacoustics of Uniform Structures in Mean Flow

Abstract

This chapter is organized as three parts: in the first part, an analytic approach is formulated to account for the effects of mean flow on sound transmission across a simply supported rectangular aeroelastic panel. The application of the convected wave equation and the displacement continuity condition at the fluid-panel interfaces ensures the exact handling of the complex aeroelastic coupling between panel vibration and fluid disturbances. To explore the mean flow effects on sound transmission, three different cases (i.e., mean flow on incident side only, on radiating side only, and on both sides) are separately considered in terms of refraction angular relations and sound transmission loss (STL) plots. Obtained results show that the influence of the incident side mean flow upon sound penetration is significantly different from that of the transmitted side mean flow. The contour plot of refraction angle versus incident angle for the case when the mean flow is on the transmitted side is just a reverse of that when the mean flow is on the incident side. The aerodynamic damping effects on the transmission of sound are well captured by plotting the STL as a function of frequency for varying Mach numbers. However, as the Mach number is increased, the coincidence dip frequency increases when the flow is on the incident side but remains unchanged when in the flow is on the radiating side. In the most general case when the fluids on both sides of the panel are convecting, the refraction angular relations are significantly different from those when the fluid on one side of the panel is moving and that on the other side is at rest.

In the second part, the transmission of external jet noise through a double-leaf skin plate of aircraft cabin fuselage in the presence of external mean flow is analytically studied. An aero-acoustic-elastic theoretical model is developed and applied to calculate the sound transmission loss (STL) versus frequency curves. Four different types of acoustic phenomenon (i.e., the mass-air-mass resonance, the standing-wave attenuation, the standing-wave resonance, and the coincidence resonance) for a flat double-leaf plate as well as the ring frequency resonance for a curved double-leaf plate are identified. Independent of the proposed theoretical model, simple closed-form formulae for the natural frequencies associated with the above acoustic phenomena are derived using physical principles. Excellent agreement between the model predictions and the closed-form formulae is achieved. Systematic parametric investigation with the model demonstrates that the presence of the mean flow as well as the sound incidence angles affects substantially the sound transmission behavior of the double-leaf structure. The influences of panel curvature together with cabin internal pressure on jet-noise transmission are also significant and should be taken into account when designing aircraft cabin fuselages.

In the third part, a theoretical model is developed to investigate the influence of external mean flow on the transmission of sound through an infinite double-leaf panel filled with porous sound absorptive materials. The sound transmission process in the porous material is modeled using the method of equivalent fluid-structure coupling conditions that are accounted for to ensure displacement continuity at fluid-structure interfaces. Analytic solutions for the sound transmission loss of the whole structure are obtained. For validation, the model predictions are compared with existing experimental results. Numerical investigations with the model are subsequently performed to quantify how a set of systematic parameters affect the sound transmission loss. It is demonstrated that the porous material affects the STL curve in terms of both the absorption effect and the damping effect. Besides, the material loss factor and the thickness of the faceplates also have an influence on the coincidence dip of the STL curve. At frequencies below the coincidence frequency, the external mean flow increases the STL values due to the added damping effect of the mean flow while shifting the coincidence frequency upward because of the refraction effect of the mean flow. In addition, the coincidence frequency decreases with increasing azimuth angle between the sound incident direction and mean flow direction.

Appendix

As stated in Sect. 2.2.7, the simple closed-form formulae, i.e., Eqs. (2.91), (2.92), (2.93), and (2.94), are applied to validate the proposed aero-acoustic-elastic theoretical model in Sect. 2.2, because these formulae are developed independent of the theoretical model. The physical nature of the four acoustic phenomena based on which the closed-form formulae are derived is presented below (Fig. 2.34).

2.1.1 Mass-Air-Mass Resonance

In the absence of the external mean flow, the formula for predicting the mass-air-mass resonance frequency of a double-leaf aeroelastic panel has been presented in Refs. [3, 5, 47, 73]. The radially outspreading bending wave in the panel caused by the incident sound leads to the highly directional sound radiation. As a result, the mass-air-mass resonance strongly depends on the incident angle. When the mass-air-mass resonance occurs, the two panels with the air cavity in between behave like a mass-spring-mass system, with the stiffness of the air cavity given by ρ ₂ c ²₂ /(H sin² φ ₁). However, due to the refraction effect of the mean flow as shown in Fig. 2.34, the incident angle φ ₁ in the presence of the mean flow will be changed to φ ₂ before the sound penetrates through the incident panel. In other words, the mean flow case is equivalent to the case when the sound is incident on the panel with angle φ ₂ in the absence of the mean flow (denoted by the dash lines in Fig. 2.34). Accordingly, the stiffness of the air cavity is changed to ρ ₂ c ²₂ /(H sin² φ ₂), and the eigenvalue equation of the equivalent mass-spring-mass vibration system becomes

Fig. 2.34

Sketch of sound transmission through double-leaf aeroelastic panel in the presence of external mean flow

$$ \left|\begin{array}{ll} \frac{\rho_2{c}_2^2}{H{ \sin}^2{\varphi}_2}-{\omega}^2{m}_1 & \frac{-{\rho}_2{c}_2^2}{H{ \sin}^2{\varphi}_2}\\ {} \frac{-{\rho}_2{c}_2^2}{H{ \sin}^2{\varphi}_2}& \frac{\rho_2{c}_2^2}{H{ \sin}^2{\varphi}_2}-{\omega}^2{m}_2 \end{array}\right|=0 $$

(2.145)

Solving Eq. (2.145) leads to the mass-air-mass resonance given in Eq. (2.91).

2.1.2 Standing-Wave Attenuation

In the presence of external mean flow, the phenomenon of standing-wave attenuation in a double-panel system is caused by the destructive interference between two meeting waves, which occurs only when the distance difference between the routes that the two intervening waves pass through is the odd number of one quarter of the incident sound wavelength. Under such conditions, the interference between the positive-gong wave and the negative-going wave tends to be destructive when the sound is transmitting through the partition, resulting in maximum sound reduction. As shown in Fig. 2.34, the plane sound ray comprised of a set of harmonic sound waves with the same vibration phase obliquely impinges on the incident panel. There indeed exists a case when one sound ray is incident on the upper panel at point A and transmits through the air cavity onto the bottom panel at point B and its reflected portion back to the upper panel at point C meets another sound ray from the external incident side, i.e., the interference between the two sound waves has occurred.

As is well known, the destructive effect between two harmonic waves occurs only when their vibration phases differ by odd numbers of π/2. In the case considered here (Fig. 2.34), the distance difference between the routes that the two intervening waves pass through is the only cause of the phase difference. For convenience, the equivalent case of sound incident with angle φ ₂ in the absence of mean flow (dash lines in Fig. 2.34) is used to represent the case of sound incident with angle φ ₁ in the presence of mean flow (solid lines in Fig. 2.34). The distance difference between the routes that the two sound waves pass through is such that

$$ \overline{AB}+\overline{BC}-\overline{CD}=\frac{2 H}{ \sin {\varphi}_2}-2 H \cot {\varphi}_2 \cos {\varphi}_2=2 H \sin {\varphi}_2 $$

(2.146)

The occurrence of the destructive effect requires that

$$ H \sin {\varphi}_2=\frac{\left(2 n-1\right)\lambda}{4}\left( n=1,2,3\dots \right) $$

(2.147)

where λ = c/f is the wavelength in air. Equation (2.92) for the standing-wave attenuation frequency follows from Eq. (2.147).

2.1.3 Standing-Wave Resonance

As the counterpart of the standing-wave attenuation, the standing-wave resonance is also caused by the interference effect between two meeting waves, which is however constructive rather than destructive. It appears when the distance difference between the routes that the two intervening waves pass through is the multiple of the half wavelength of the incidence sound wave. In such cases, the constructive interference between the positive-going and negative-going waves leads to an enhanced transmission through the double-panel partition. The condition for the appearance of the standing-wave resonance is given by

$$ H \sin {\varphi}_2=\frac{n\lambda}{2}\left( n=1,2,3\dots \right) $$

(2.148)

With the relation λ = c/f, Eq. (2.148) can be readily converted to Eq. (2.93).

2.1.4 Coincidence Resonance

Similar to the standing-wave attenuation and resonance, the coincidence resonance is also an interference effect between two intervening waves by physical nature. The difference between the two different types of acoustic phenomenon is that the coincidence resonance is caused by wave interference between the sound wave and the panel flexural bending wave. As shown in Fig. 2.34, there exists a situation when the panel flexural bending wave at point A (excited by the incident sound ray at this point) rapidly outspreads to point C, where it matches another sound ray from the external incident side. The premise for this matching is that the flexural bending wave in the aeroelastic panel propagates faster than sound propagation in air, requiring that the former transmits a larger distance than the latter during the same time period (i.e., $\overline{AC}>\overline{DC}$ in Fig. 2.34), and hence

$$ \frac{\overline{AC}}{c_t}=\frac{\overline{DC}}{c} $$

(2.149)

This premise is in general satisfied in practice. Consequently, the constructive interference between the two waves results in the coincidence resonance. From the governing equation of panel vibration, we have

$$ {c}_t=\sqrt[4]{\frac{D{\omega}^2}{m}} $$

(2.150)

Substitution of Eq. (55) and the relation $\overline{DC}=\overline{AC}\cdot \cos {\varphi}_2$ into Eq. (2.149) leads to Eq. (2.94) for the coincidence resonance frequency.

Note that the coincidence resonance will occur in the bottom panel as well when the upper panel generates coincidence resonance. In other words, the incident sound will be enhanced twice, initially by the upper panel and followed by the bottom panel, when the condition for the occurrence of the coincidence resonance is satisfied.