Acoustic transmission loss and noise from Kevlar wind tunnel walls

Abstract

In this paper we will develop a model for the acoustic transmission loss and self-noise generated by a Kevlar wind tunnel wall. It is shown that the porosity of the fabric is the most important controlling factor of the transmission loss, and the effect of wind tunnel flow speed is to increase the losses, as observed in experiments. In addition, a model is developed for the weave noise generated by a Kevlar wind tunnel wall, which is found to be caused by the pumping of the fluid through the pores in the Kevlar and depends on their open area ratio. The mechanism for this sound generation is similar to the roughness noise mechanism for a turbulent boundary layer in that the pore spacing couples with the small wavelength disturbances in the boundary layer to cause acoustic radiation at the sum and difference frequencies.

Keywords

Aeroacoustics wind tunnels aeroacoustic testing kevlar wind tunnel walls acoustic transmission loss

Introduction

Wind tunnels with Kevlar walls¹ have been developed to facilitate measurements of aeroacoustics noise sources, such as the noise from wind turbine blades or the broadband noise from propellers and rotors. The Kevlar walls have been shown experimentally^1,2 to have very low acoustic transmission loss, while maintaining a controlled aerodynamic flow over the system being tested. The most commonly used Kevlar fabric is style K120 consisting of K49 yarns, which is the lightest available on the market. Namely, it has a mass per unit area of 58 gm/m², while its tensile strength exceeds that of mild steel. The acoustic transmission loss of K120 has been found to be on the order of 1–2 dB at 10 kHz when there is no flow in the wind tunnel. The transmission loss increases with wind tunnel flow speed, and at 72 m/s the measurements¹ show a transmission loss of 5 dB at 10 kHz. These are acceptable corrections and enable aeroacoustic testing on a wide range of devices at realistic flow speeds. Remarkably, however, there is no analytical model to substantiate these transmission loss measurements. For the no-flow case one might expect that the transmission loss could be calculated based on the mass per unit area of the fabric, but this provides a transmission loss estimate of 13 dB at 10kHz (see The Introduction), which is an order of magnitude larger than the measurements suggest. The Kevlar, however, is porous, and so a porous wall model is required. In this paper we will develop such a model, including the effect of wind tunnel flow speed, and show that it is the porosity of the wall that controls its acoustic properties.

Kevlar wind tunnel walls have also been found to generate their own self noise at high frequencies when compared to open jet facilities. Bahr and Hutcheson³ compared background noise levels of the NASA Langley Quiet Flow Facility with and without Kevlar walls and found a high frequency spectral hump at high frequencies (20–60 kHz) when Kevlar walls were present and this noise was found to increase in level with flow speed. Similar results have been found in other facilities, for example the Anechoic Wall Jet Tunnel at Virginia Tech showed increases in tunnel noise when a Kevlar panel was mounted in the wall, as shown in Figure 1. This is undesirable for aeroacoustic testing and was attributed to the interaction of the turbulent boundary on the Kevlar wall with the roughness of the fabric surface.

A limited number of experimental studies are available in the literature that focus on the aeroacoustic properties of Kevlar or porous fabrics. A series of experiments in a wall jet facility were carried out by Alexander and Devenport⁴ on different fabrics, and they also found weave noise generated by the fabric. They argued that the weave noise was related to the scattering of turbulent wall pressure fluctuations into acoustic waves by fabric roughness, rather than the perforated plate model developed by Nelson.⁵ In a more recent study, Szőke et al.⁶ assessed the self-noise generation and transmission loss of porous fabrics, including commonly used Kevlar, custom Kevlar, and glass fiber fabrics. They reported that the transmission loss of the fabrics was primarily determined by their open area ratio (OAR). The self-noise generated by the fabrics was related to the weave spacing (thread per inch, TPI) and the OAR.

Sound generation by a turbulent boundary layer over a porous surface was first discussed by Ffowcs-Williams.⁷ He considered the flow over an acoustic liner that consisted of rigid or compliant surfaces with uniformly spaced holes covering a sound absorbing material. For a rigid surface it was shown that the boundary layer turbulence could be scattered into acoustic waves that scaled with the fourth power of the flow speed. However, for a completely compliant surface the scaling was dependent on the sixth power of the flow speed, typical of a dipole sound source.

Howe⁸ considered a flexible perforated plate with a specified bending stiffness and mass per unit area, and a uniform distribution of holes in the plate. He equated the pressure jump across the plate to the sum of the mechanical impedance of the plate times the plate normal velocity, and an attached mass loading that depended on the flow speed through the holes. The relative importance of these two terms was specified by the open area ratio. The leading order assumption of this analysis was that the holes in the plate were sufficiently far apart that the flow through each hole could be modelled as if it was a single hole in isolation. The interaction between the flow in each hole was not considered.

The model developed by Howe was later used by Jaworski and Peake⁹ to analyze the trailing edge noise from a semi-infinite plate with a porous surface. They determined that the porosity affected the trailing edge noise differently at different non-dimensional frequencies and specified their results in terms of a frequency dependent Mach number scaling. Hajian and Jaworski¹⁰ applied a similar model to the flutter conditions of a tensioned membrane with a distribution of vorticity. They did not consider each pore independently and included a continuous distribution of porosity. However, they did not include the mechanical impedance of the membrane in their analysis.

The focus of this paper is the sound transmission through, and sound generation by, wind tunnel walls that are made of a porous Kevlar fabric. In this case the fabric is inhomogeneous and anisotropic but has continuous properties. This distinguishes it from the liner model in which the porosity was introduced by a uniform array of holes in an impervious plate. For the fabric, the seepage flow will be a continuous function of position and will couple with the acoustic waves directly rather than as the superposition of a discrete array of monopole sources. In the low frequency limit this may be a small distinction, but at high frequencies, where the scales of the boundary layer turbulence are similar to the weave dimension, the detailed modeling of the weave will be important, and so this feature needs to be retained in the analysis.

In the following sections we will first develop a simple model for the motion of the Kevlar wall when excited by an acoustic or hydrodynamic wave, and then derive an expression for the acoustic transmission loss through the wall. We will then extend the model to consider the self-noise generated by the wall and obtain a prediction of the wall self-noise caused by a turbulent boundary layer on its surface. This has the same characteristics as the measurements made by Bahr and Hutcheson ³ and its relationship to the wall porosity will be defined.

Acoustic transmission loss

Introduction

The sound transmission through a partition as shown in Figure 2, is well understood and defined in terms of the transmission coefficient T_L, which is the ratio of the transmitted wave amplitude to the incident wave amplitude. It depends on the mechanical impedance of the wall, Z_W, which specifies the ratio of the pressure jump across the wall to normal velocity of the wall displacement u₂. Extensions to the case of a perforated screen, with either a bias flow or a grazing flow are given by Howe⁸ who gives the transmission loss for an acoustic wave incident at the angle θ as

\begin{array}{l} T_{L} = \frac{2 Z_{e}}{2 Z_{e} + \frac{Z_{W} \cos θ}{ρ_{o} c_{o}}} Z_{W} = - i ω m (1 - (\frac{c_{s}^{2}}{c_{o}^{2}}) \sin^{2} θ) \\ Z_{e} = 1 + 2 N R K_{R} (\frac{1 + i Z_{W}}{2 ρ_{o} ω R}) \end{array}

(1)

where Z_W is the wall impedance, θ is the angle of incidence of the incoming wave and ρ_oc_o is the acoustic impedance of the medium, N is the number of pores per unit area, c_s is the membrane wave speed, m is the membrane mass per unit area, R is the pore radius and K_R is the Rayleigh conductivity (see Howe⁸).

For the case of a non-porous Kevlar wall, which can be modeled as a lightweight tensioned membrane, the wall impedance Z_W only includes the mechanical impedance of the membrane which can be approximated by Z_W =-iωm and Z_e=1. In that case, for a typical mass per unit area of 58 gm/m² the transmission coefficient would be T_L=(1+i4.39)⁻¹ or a transmission loss of 13dB. The effect of wall porosity is determined by the Rayleigh conductivity K_R which for the case when there is no flow in the wind tunnel is given by 2R. Figure 3 shows a comparison of the transmission loss for the no flow case for a typical Kevlar wall and compares the non-porous case with the experimentally obtained empirical model for the Kevlar transmission loss given in Devenport et al.¹ These calculations were done using a mass per unit area of 58 g/m², a weave spacing of 34 threads per inch and two different pore sizes, corresponding to a pore diameter of either 66% and 20% of the measured pore diameter of 0.17mm as given in Devenport et al.¹ Note that the less porous wall has a higher transmission loss than measured, and the effect of the porosity is to reduce the loss. Also, if the effective pore diameter is taken as 66% the measured pore diameter then a perfect fit to the measured transmission loss is obtained using Howe’s model.

Howe discusses the effect of a grazing flow on the membrane and gives a formula for the corrections to the Rayleigh conductivity to account for the grazing flow. His model focused on the generation of vorticity by the flow though the pores and the effect of the mean grazing flow on this characteristic. His results are determined by the Strouhal number ωR/U where U is the flow speed. For the case of a pore in a Kevlar sheet this parameter is typically 0.35 for a 30 m/s flow at 10 kHz and so is relatively small. Howe’s results are presented in terms of the non-dimensional parameters Γ_R and Δ_R so that the corrected conductivity is given as K_R=2R(Γ_R+iΔ_R). Using the numerical results given by Howe and the computed the transmission loss far exceeds the measured transmission loss in the case of a wind tunnel flow. One possible explanation for this is that in Howe’s model the edges of the holes in the perforated plate were sharp and assumed to shed unsteady vorticity. The flow through a porous fabric will not encounter the same edge conditions as the holes in a plate and so a different behavior is to be expected and vortex shedding is less likely to occur. For this reason, we will reconsider the problem and derive the acoustic transmission loss in terms of a porosity parameter β that can be used to specify the behavior of a porous fabric as distinct from the modification of the formulas derived by Howe for porous plates.

The hydrodynamic and acoustic fields

Consider an acoustic plane wave that is incident on a Kevlar wall from inside a wind tunnel where the mean flow speed is U in the x₁ direction as shown in Figure 2. In this section we will follow the approach given by Morse and Ingard¹¹, with the exception that we will not restrict the waves inside the tunnel to be propagating acoustic waves, and allow them to have evanescent decay away from the wall. This extension allows for the specification of boundary layer pressure fluctuations or hydrodynamic waves caused by turbulent flow within the framework of the acoustic wave equation.

In the tunnel, the acoustic waves satisfy the convected form of the acoustic wave equation

\frac{1}{c_{o}^{2}} \frac{D_{o}^{2} p_{s}}{D t^{2}} - \nabla^{2} p_{s} = 0

(2)

If the incident wave is of amplitude A and the reflected wave is of amplitude B then the acoustic pressure in the wind tunnel p_s will be of the form

p_{s} = p_{i} + p_{r} p_{i} = A e^{- i ω t + i k_{1} x_{1} + i k_{2} x_{2}} p_{r} = B e^{- i ω t + i k_{1} x_{1} - i k_{2} x_{2}}

(3)

Where the characteristic equation for the wavenumbers is obtained from the convected wave equation as

{(\frac{ω}{c_{o} - k_{1} M})}^{2} - k_{1}^{2} - k_{2}^{2} = 0

(4)

In the medium outside the wind tunnel the acoustic wave satisfies the wave equation for a stationary medium and the pressure will be given by a wave of amplitude C defined so

p_{t} = C e^{- i ω t + i k_{1} x_{1} + i k_{2}^{(t)} x_{2}}

(5)

Outside the tunnel in the stationary medium the waves propagate as acoustic waves, at an angle of transmission θ_t and so k₁=(ω/c_o) sinθ_t and k₂^(t)=(ω/c_o) cosθ_t. Given these relationships we find that the characteristic equation for the waves in the tunnel yields the wall normal wavenumber

k_{2} = (\frac{ω}{c_{o}}) \sqrt{{(1 - M \sin θ_{t})}^{2} - \sin^{2} θ_{t}}

(6)

This shows that at large angles of transmission θ_t the waves outside the tunnel are generated by evanescent waves in the tunnel because k₂ is imaginary. However, at normal incidence where θ_t=0, we see that k₁=0 and k₂=k₂^(t)=ω/c_o.

To determine the transmission coefficient, we must match the acoustic waves to the displacement of the wall and any flow through the wall due to the porosity of the Kevlar. Before deriving the transmission coefficient, we will first consider these two properties in more detail.

The wall motion

We will model the surface velocity of the fluid over the Kevlar sheet as the sum of two components u₂=u_s+u_p. The first, u_s, is due to the motion of the surface that is driven by the pressure jump across the wall, and the second part u_p is the flow through the wall that only occurs because the wall is porous.

For a tensioned membrane, whose motion is defined by the displacement ζ in the x₂ direction, the motion of the membrane is given by⁸

(\frac{\frac{\partial T}{\partial s}}{m}) \nabla^{2} ζ - \frac{\partial^{2} ζ}{\partial t^{2}} = - \frac{Δ p}{m}

(7)

where ∂T/∂s is the tension per unit length in the membrane and m is the mass per unit area of the membrane, and Δp is the pressure jump across the membrane.

The motion of the membrane depends on the incoming wave and so, to be consistent with surface pressure p_s, which is equal to p_i+p_r on the surface, we define the incident wave to cause a motion the type $ζ = \hat{ζ} \exp (- i ω t + i k_{1} x_{1})$ and define a wavenumber dependent impedance. However, the velocity of the fluid next to the surface is affected by the horizontal flow in the wind tunnel (see Figure 2) and so the relationship between the normal velocity and the displacement of the Kevlar will be different on each side of the surface. On the upper surface (the flow side), the normal velocity depends on $\frac{D ζ}{D t} = - (i ω - k_{1} U) \hat{ζ}$ and on the lower surface (stationary fluid side) on $\frac{\partial ζ}{\partial t} = - i ω \hat{ζ}$ . Hence, we define the flow normal to the wall in terms of a mechanical impedance Z as

{\hat{u}}_{s}^{(-)} = - i ω \hat{ζ} = \frac{({\hat{p}}_{s} - {\hat{p}}_{t})}{Z} {\hat{u}}_{s}^{(+)} = (1 - M \sin θ_{t}) {\hat{u}}_{s}^{(-)}

(8)

where

{\hat{p}}_{t}

and

{\hat{p}}_{s}

are the harmonic pressure wave amplitudes on the outer and inner surfaces of the Kevlar wall, the +/− superscript refers to the flow side and stationary fluid side surfaces in Figure 2, and the harmonic time and space dependence has been suppressed. It is assumed here that any inhomogeneity of the mass of the membrane or anisotropy of the tension is of a much smaller scale than the wavelength of the membrane motion so the average mass per unit area may be used in the equation of motion. This implies that the Kevlar weave dimension, which is typically less than a millimeter, is small compared to the acoustic wavelength, which may be a limitation at very high frequencies.

The equation for membrane motion is put in terms of the surface velocity and is then determined from equation (7), in the wavenumber domain, as

(c_{s}^{2} k_{1}^{2} - ω^{2}) {\hat{u}}_{s}^{(-)} = - \frac{i ω ({\hat{p}}_{s} - {\hat{p}}_{t})}{m} c_{s} = \sqrt{\frac{(\frac{\partial T}{\partial s})}{m}}

(9)

where c_s is the speed of propagation of the waves in the membrane. The mechanical impedance is then given by

Z = - i ω m (1 - \frac{c_{s}^{2} k_{1}^{2}}{ω^{2}})

(10)

For a typical tension of 1500 N/m and mass per unit area of 58 gm/m² we find that c_s=160 m/s, and so the effect of tension on the wall impedance cannot be ignored for waves at high angles of transmission where k₁=(ω/c_o)sinθ_t, but is negligible for normal incident waves for which k₁=0.

The effective impedance of the porous wall

The effect of the porosity is to allow flow through the wall that must be combined with the wall motion to obtain the total effective impedance. However, the porosity of the Kevlar will not be uniform as the leakage though the membrane only occurs at the center of the overlapping threads in the weave. Howe⁸ considered the flow through a perforated plate with equally spaced circular holes but assumed that each hole was sufficiently far apart that it acted independently. For the case of a porous fabric, the hole size and spacing is of the same order of magnitude and so this assumption cannot be made. The flow through the weave must therefore be modulated by a flow pattern that will be modeled using a porosity function such that

u_{p} (x_{1}, x_{3}, t) = {\hat{u}}_{p} e^{- i ω t + i k_{1} x_{1} + i k_{3} x_{3}} f_{o} (x_{1}, x_{3})

(11)

where f_o(x₁,x₃) is a non-dimensional function describing the weave porosity which has a unit average value, so the average unsteady flow through the wall can be defined as

{\hat{u}}_{p} e^{- i ω t + i k_{1} x_{1} + i k_{3} x_{3}}

(12)

As the next step we define a non-dimensional flow conductivity parameter β so

{\hat{u}}_{p} = (\frac{β}{ρ_{o} c_{o}}) ({\hat{p}}_{s} - {\hat{p}}_{t})

(13)

This can also be directly related to the flow (or Rayleigh) conductivity per unit area of the Kevlar K_R, [7, p354] which gives

{\hat{u}}_{p} = (\frac{K_{R}}{i ω ρ_{o} S_{o}}) ({\hat{p}}_{s} - {\hat{p}}_{t})

(14)

where u_pS_o is the volume flowrate through the aperture. Since u_p is the average flow across the surface it follows that S_o is the surface area of the fabric per pore.

The total effective impedance of the wall then is defined by considering the fluid velocity at the wall, averaged over the weave, as the sum of the wall motion and the flow through the wall. The relationship between this velocity and the pressure jump across the wall is then

{\hat{u}}_{2}^{(-)} = {\tilde{u}}_{s}^{(-)} + {\hat{u}}_{p} = (\frac{1}{Z} + \frac{β}{ρ_{o} c_{o}}) ({\hat{p}}_{s} - {\hat{p}}_{t}) {\hat{u}}_{2}^{(+)} = (\frac{1 - M \sin θ_{t}}{Z} + \frac{β}{ρ_{o} c_{o}}) ({\hat{p}}_{s} - {\hat{p}}_{t})

(15)

The acoustic transmission loss

To calculate the transmission coefficient for the acoustic wave we use the acoustic momentum equation on each side of the surface to relate the pressure and velocity components, and combine the results with equation (15) to find the ratio of the transmitted to incident wave amplitudes C/A=T_L. The momentum equations are

ρ_{o} \frac{D_{o} u_{2}^{(+)}}{D t} = - \frac{\partial p_{s}}{\partial x_{2}}

(16)

ρ_{o} \frac{\partial u_{2}^{(-)}}{\partial t} = - \frac{\partial p_{t}}{\partial x_{2}}

(17)

and so

i ω ρ_{o} (1 - M \sin θ_{t}) {\hat{u}}_{2}^{(+)} = i k_{2} (A - B) i ω ρ_{o} {\hat{u}}_{2}^{(-)} = i k_{2}^{(t)} C

(18)

which we will rearrange as

{\hat{u}}_{2}^{(+)} = \frac{(A - B) α \cos θ_{t}}{ρ_{o} c_{o}} {\hat{u}}_{2}^{(-)} = \frac{C \cos θ_{t}}{ρ_{o} c_{o}} α = \frac{k_{2}}{k_{2}^{(t)} (1 - M \sin θ_{t})}

(19)

And since ${\hat{p}}_{s} = A + B a n d {\hat{p}}_{t} = C$ equation (9) is

{\hat{u}}_{2}^{(+)} = (\frac{1 - M \sin θ_{t}}{Z} + \frac{β}{ρ_{o} c_{o}}) (A + B - C) {\hat{u}}_{2}^{(-)} = (\frac{1}{Z} + \frac{β}{ρ_{o} c_{o}}) (A + B - C)

(20)

These equations are solved by specifying the coefficients K₊ and K-, where

\begin{array}{l} K_{+} (A - B) = A + B - C K_{-} C = A + B - C \\ K_{+} = \frac{α \cos θ_{t}}{ρ_{o} c_{o}} {(\frac{1 - M \sin θ_{t}}{Z} + \frac{β}{ρ_{o} c_{o}})}^{- 1} \\ K_{-} = \frac{Z_{W} \cos θ_{t}}{ρ_{o} c_{o}} \frac{1}{Z_{W}} = (\frac{1}{Z} + \frac{β}{ρ_{o} c_{o}}) \end{array}

(21)

Hence, we obtain the acoustic transmission coefficient as

T_{L} = \frac{2}{T_{L} = \frac{2}{1 + K_{-} / K_{+} + K_{-}}}

(22)

At zero angle of incidence K_-=K₊, the mechanical impedance becomes mass controlled, so Z=-iωm and the waves are not refracted, and we obtain a relatively simple formula for the transmission coefficient

T_{L} = \frac{1 - i ω m β / ρ_{o} c_{o}}{1 - \frac{i ω m}{2 ρ_{o} c_{o}} (2 β + 1)}

(23)

From this we see that the effect of the porosity, which depends on β, is to effectively reduce the transmission loss through the wall, and at high frequencies where ωm/ρ_oc_o>>1, we find that

T_{L} = \frac{2 β}{2 β + 1}

(24)

The acoustic properties of the wall are therefore strongly dependent on the flow conductivity parameter β and to understand this further we need to investigate its characteristics in more detail.

The flow though the wall

To obtain the conductivity in a porous plate Howe⁸ considered the Rayleigh conductivity K_R which is related to the non-dimensional conductivity β as

β = (\frac{K_{R} c_{o}}{i ω S_{o}})

(25)

In the ideal case it is shown in⁸ that K_R equals the pore diameter, and this obviously will be less than the weave spacing d. If we define the pore diameter as d_o and the area of fabric per pore as S_o=d² we obtain

β = (\frac{c_{o}}{i ω d}) Γ_{R}

(26)

Where Γ_R=d_o/d is the ratio of the pore diameter to the thread spacing. In most applications the acoustic wavelength is much larger than the pore diameter and so ωd/c_o<<1 so β>>1 which implies that the transmission coefficient T_L=1, or C=A, so there is complete acoustic transmission. However, we still need to define the pore diameter and this can only be inferred from measurements, and may vary from sample to sample.¹

It could also be argued that the flow through the Kevlar is dominated by viscous effects and so we also need to investigate the implications of such a model. For the viscous case the pressure drop across the fabric is based on the theory for flow through porous media (see¹² for a review). This is based on Darcy’s equation and gives the flow speed through the pores in the material as

{\hat{u}}_{p} = \frac{K ({\hat{p}}_{s} - {\hat{p}}_{t})}{μ L}

(27)

where K is the permeability (with units of m²), L is the length of the pore, and will be taken as equal to the pore diameter, and μ is the dynamic viscosity. The Reynolds number of the flow through the pores for a sound wave with an SPL of 90 dB re 20 μPa and a pore diameter of ∼1mm is ∼0.1 and so the acoustic wave is within the range of flow speeds where this law applies (Yazdchi et al.¹²). The Carman model for the permeability¹² gives

K = \frac{ε D_{h}^{2}}{32}

(28)

where ε =1−ϕ is the porosity, with ϕ equal to the solid volume fraction of the porous material, which implies that ε =(d_o/d)². Also, D_h is the hydraulic diameter, which can be approximated¹² by

D_{h} = \frac{ε d}{1 - ε}

(29)

where d is the weave spacing. Given these assumptions, and taking L=d, we then estimate β as

β = \frac{ρ_{o} c_{o} d ε^{3}}{32 μ {(1 - ε)}^{2}}

(30)

The parameter β is seen to be a Reynolds number based on the weave spacing and the speed of sound. For weave spacing of the order of 1 mm we find β =708ε³/(1-ε)². Compete transmission is then only expected for highly porous materials, and as the porosity is reduced then more transmission loss is expected.

For the case of no flow we can compare the predictions using the formulas outlined above to the empirical formulas for transmission loss given in Glegg and Devenport.¹³ The calculations are carried out for a weave of 34 threads per inch, with a mass per unit area of 58 gm/m², with a tension of 1500 N/m. Figure 4 shows transmission loss calculations based on a viscous model defined by equation (30), for different values of the porosity and an inviscid model based on equation (26). Note how the porosity increases the attenuation through the wall for the viscous model, and that this model greatly exceeds the measured transmission loss. However, for the inviscid model with β defined by equation (26) an almost exact agreement with the measurements is obtained by setting the pore diameter to weave spacing ratio Γ_R=0.17. The actual pore size for this fabric was given in^1,12 as 0.17 mm for a thread spacing of 0.74 mm, so the effective pore diameter is 75% of the actual pore size. We conclude from this model that increasing the pore size reduces the transmission loss as it increases the value of β. Consequently, a less tight weave will be more acoustically transparent. Similarly, Szőke et al.^6,11,15–18 found the open area ratio as the primary factor determining the transmission loss of Kevlar fabrics.

The effect of wind tunnel flow

To account for the effect of mean flow in the wind tunnel we need to account for the static pressure jump across the wall as this sets up a mean flow, or bias flow, through the Kevlar that can be much larger than the acoustically induced flow. The steady flow through a Kevlar wall has been measured in static tests and the relationship between the flow and the pressure drop across the wall is given by the empirical formula¹

U_{p} = k_{c} \sqrt{Δ p} {(\frac{Δ p}{Δ p + k_{p}})}^{μ_{o}}

(31)

with k_c=0.0625 m/(s√Pa), k_p=50 Pa and μ_ο=0.073 for an open area ratio

{(d_{o} / d)}^{2}

of about 7%. (Note there is an error in ¹ that gives k_c as 10 times the correct value given here). This relationship is only a minor modification of Bernoulli’s equation for orifice type flow through the pores. Ignoring the small corrections represented by k_p and μ_o, k_c can be roughly estimated as

{(d_{o} / d)}^{2} C_{c} \sqrt{\frac{2}{ρ_{o}}}

where

C_{c}

=0.61 is the vena contracta ratio.¹⁸ The pressure jump, Δp, is determined by the difference between the aerodynamic pressure exerted on the inside of the wall and the ambient pressure outside, which in the absence of any significant model blockage would be 0–2% of the free stream dynamic pressure ρ_οU²/2. For a high lift model, however, pressures can reach 10 to 20 times this. Thus for an empty tunnel flow speed of 30 m/s the flow speed U_p through the wall is of order 0.1m/s rising for 0.5 m/s for large blockage. These values far exceed any acoustic velocity and exceeds any velocity likely to be induced by boundary layer pressure fluctuations. Consequently, for the higher tunnel flow speeds we need to consider the unsteady part of the through flow as a perturbation to the mean flow through the wall.

If the net flow through the membrane is $U_{p} + {\tilde{u}}_{p} \exp (- i ω t)$ , then for small perturbations of pressure, we can use the expansion

U_{p} + {\hat{u}}_{p} \exp (- i ω t) \approx U_{p} + \frac{d U_{p}}{d (Δ p)} Δ \hat{p} \exp (- i ω t)

(32)

So, from equation (13), assuming that μ_ο<<1

\frac{β}{ρ_{o} c_{o}} = \frac{d U_{p}}{d (Δ p)} \approx (\frac{k_{c}}{2 \sqrt{Δ p}}) {(\frac{Δ p}{Δ p + k_{p}})}^{μ_{o}}

(33)

The implication is that when $Δ p = α_{W}^{2} ρ_{o} U^{2} / 2$

\frac{β}{ρ_{o} c_{o}} \approx \frac{k_{c}}{α_{W} U \sqrt{2 ρ_{o}}}

(34)

For an empty wind tunnel, we can take $α_{W}^{2} = 0.01$ and so

β \approx \frac{ρ_{o} c_{o} k_{c}}{0.1 U \sqrt{2 ρ_{o}}} = 10 (\frac{k_{c}}{M} \sqrt{\frac{ρ_{o}}{2}}) = (\frac{0.48}{M})

(35)

where M is the Mach number of the wind tunnel flow speed.

The effect of flow speed on the transmission loss is shown in Figure 5 using the model specified by equation (34). At low Mach numbers these results show an increase in Transmission Loss with flow speed which is similar to those measured, but at higher flow speeds the measured losses are greater than predicted by this formula, and smaller values of β are needed at high wind tunnel speeds. At the highest speed shown reducing the flow conductivity to 25% of its predicted value gives a fit to within a dB of the measured data (Figure 6). This is consistent with the discussion given by Howe⁸ which suggests that a grazing flow will reduce the conductivity for small pores due to the convection of vorticity across the pore interface with the flow, but it is inconsistent with the discussion in¹ which suggests that for a higher pressure drop the pores in the fabric open up and the conductivity is increased. To address this problem an elastic pore response is required, and this does not appear to have been considered in the literature.

Roughness and weave noise from a Kevlar window in a wind tunnel

In previous theories on roughness noise the source of the radiated sound has been modeled by the interaction of the roughness elements on a rigid impermeable wall with the surface pressure fluctuations caused by a turbulent boundary layer. For flow over a Kevlar wall the theory must be modified to include the effect of wall flexibility and permeability. Furthermore, there will be sound transmitted through the wall and, since this is of most interest in wind tunnel applications, the roughness noise transmitted through the wall needs to be considered. In addition, there is flow through the wall which can be characterized by an array of equally spaced jets of fluid (see Figure 7) and this will cause additional noise, which we will refer to as weave noise.

Surface generated noise

To analyze this problem, we must specifically take into account the motion of the surface and so, rather than start with Curle’s equation as was done in¹³ we will base the analysis on the Ffowcs Williams and Hawkings Equation using the notation given in.¹⁴

p' (x, t) = \int_{- T}^{T} \int_{S (τ)} ((p_{i j} + ρ v_{i} (v_{j} - V_{j})) \frac{\partial G}{\partial y_{i}} n_{j} - \frac{\partial G}{\partial τ} (ρ v_{j} - ρ' V_{j}) n_{j}) d S (y) d τ + \int_{- T}^{T} \int_{V (τ)} (T_{i j} \frac{\partial^{2} G}{\partial y_{i} \partial y_{j}}) d V (y) d τ

(36)

where p_ij is the compressive stress tensor, v_i is the velocity of the fluid, V_j is the velocity of the surface, G is a tailored Greens function for an observer at x and time t for a source at y and time τ. The coordinates are given by (y₁,y₂,y₃) with y₁ in the direction of the flow in the wind tunnel, which lies in the region y₂>0, and the wall lies in the plane surface f(y₁,y₂,y₃,t)=0. The density is defined as ρ = ρ_ο+ρ^′ but in acoustic applications or low Mach number flows ρ^′<<ρ_ο and so we can replace ρ by ρ_o. Furthermore, if the quadrupole source terms are limited to a thin layer whose thickness is small compared to the acoustic wavelength then the volume integral can be ignored in comparison to the surface source terms. We can also ignore the viscous terms so that the compressive stress tensor is p_ij=pδ_ij where p is the surface pressure.

For a rigid impenetrable surface, the remaining terms that depend on the surface and fluid velocity are zero because v_jn_j=0 and V_j=0. However, for the Kevlar wall these terms may be important. We can however make the approximation that

ρ v_{i} (v_{j} - V) \frac{\partial G}{\partial y_{i}} n_{j} < < \frac{\partial G}{\partial τ} (ρ v_{j} - ρ' V_{j}) n_{j}

(37)

because

| \frac{\partial G}{\partial y_{i}} n_{j} | < \frac{1}{c_{o}} | \frac{\partial G}{\partial τ} |

(38)

where c_o is the speed of sound,¹⁴ and the flow speeds are everywhere much less than c_o. The acoustic far field sound is then given by

p' (x, t) = \int_{- T}^{T} \int_{S} (p \frac{\partial G}{\partial y_{i}} n_{i} - ρ_{o} v_{j} n_{j} \frac{\partial G}{\partial τ}) d S (y) d τ

(39)

This equation applies on both sides of the surface. For sound radiation outside the wind tunnel, we can assume that the surface pressure is p=p_t, while for the sound field inside the wind tunnel we can assume that the surface pressure is p=p_s. The second term in equation (39) represents the sound generated by the surface motion and the leakage flow through the wall, which is expected to be the primary source of weave noise.

We will model the surface velocity term as the sum of two components v₂=u_s+u_p as was done for the acoustic transmission loss. The first, u_s, is due to the motion of the surface normal to the wall that is driven by the pressure jump across the wall. In addition to the motion of the surface the porosity of the membrane allows flow through the surface.

An important simplification is obtained by making use of a tailored Greens function that matches the non-penetration boundary condition on the surface y₂=0. To formalize this, we first note that the surface integral may be projected onto the plane y₂=0 by using the transformation,^13,14 that

n_{j} d S = \frac{\partial f}{\partial y_{j}} d y_{1} d y_{3}

(40)

where the moving surface is defined by f=0. Then

p' (x, t) = \int_{- T}^{T} \int_{- R_{1}}^{R_{1}} \int_{- R_{3}}^{R_{3}} (p \frac{\partial G}{\partial y_{i}} \frac{\partial f}{\partial y_{i}} - ρ_{o} v_{i} \frac{\partial f}{\partial y_{i}} \frac{\partial G}{\partial τ}) d y_{1} d y_{3} d τ

(41)

The definition of the surface depends on the displacement of the surface ζ(y₁,y₃,τ) and the surface roughness. The roughness height is specified by the function ξ(y₁,y₃) and so we can define the surface as

f (y_{1}, y_{3}, τ) = y_{2} - ξ (y_{1}, y_{3}) - ζ (y_{1}, y_{3}, τ)

(42)

and so

\frac{\partial f}{\partial y_{1}} = - \frac{\partial ξ}{\partial y_{1}} - \frac{\partial ζ}{\partial y_{1}} \frac{\partial f}{\partial y_{2}} = 1 \frac{\partial f}{\partial y_{3}} = - \frac{\partial ξ}{\partial y_{3}} - \frac{\partial ζ}{\partial y_{3}}

(43)

In general, the slope of the roughness $\partial ξ / \partial y_{i}$ will be of order one or larger, in fact it could be infinite.¹¹ In contrast the slope of the surface displacement will be small. For example, for harmonic motion of the membrane $| \partial ζ / \partial y_{1} | = k_{1} u_{s} / ω$ . For an acoustic wave k₁/ω is 1/c_o and so this term is very small. For a hydrodynamic wave formed in the boundary layer on the inside of the wall we find that k₁/ω=1/U_c where U_c is the convection velocity of the turbulent eddies in the boundary layer. Since we can assume that u_s is less than the turbulent velocity in the boundary layer, and that is much less than the convection velocity, it follows that $| \partial ζ / \partial y_{1} | < < 1$ and so has a minimal impact on the gradient of the function f. Hence the flow on the Kevlar is dominated by the flow in the y₂ direction, caused by through flow and displacements, and the flow parallel to the local surface. Since the latter has no velocity normal to the surface it does not contribute to the surface integral and, within the integrand, we can specify the pumping motion and the wall displacement velocity as being only in the direction normal to the y₂ plane, so

v_{i} \frac{\partial f}{\partial y_{i}} = v_{2}

(44)

To investigate this result, we take the Fourier transform with respect to time¹⁴ of equation (41), so, in terms of the pressure outside the tunnel, $\tilde{p} (x, ω)$ outside the tunnel and the surface pressure ${\tilde{p}}_{t} (y, ω)$ on the outer wall

\tilde{p} (x, ω) = \int_{- R_{1}}^{R_{1}} \int_{- R_{3}}^{R_{3}} ({\tilde{p}}_{t} \frac{\partial \tilde{G}}{\partial y_{i}} \frac{\partial f}{\partial y_{i}} + i ω \tilde{G} ρ_{o} v_{2}) d y_{1} d y_{3}

(45)

As discussed in,^13,14 there is some flexibility in choosing the Greens function in this equation. For example, the free field Greens function could be used so both terms would contribute. Alternatively, if it were a flat surface we could choose $\partial \tilde{G} / \partial y_{2} = 0$ on the surface and so the acoustic field is determined by the velocity v₂ only and the pressure term is eliminated. Since the surface is not flat there will be additional sound caused by the pressure fluctuations on the roughness, which is referred to as roughness noise, Glegg and Devenport.¹³ So, to include the roughness noise we introduce the tailored Greens function for which $\partial \tilde{G} / \partial y_{2} = 0$ on y₂=0, and using the far field approximation, we can define this as

\tilde{G} (x | y) = \frac{\cos (\frac{k x_{2} y_{2}}{r})}{2 π r} e^{i k r - i k x_{1} y_{1} / r - i k x_{3} y_{3} / r}

(46)

The rough surface is defined on the outside of the tunnel by y₂=ξ where the roughness height h>|ξ|. Then by ignoring terms of order kh, which allows us to set y₂=0 in the tailored Greens function, we obtain

\tilde{p} (x, ω) = \frac{e^{i k r}}{2 π r} \int_{- R_{1}}^{R_{1}} \int_{- R_{3}}^{R_{3}} ({\tilde{p}}_{t} {[\frac{i k x_{j}}{r} \frac{\partial ξ}{\partial y_{j}}]}_{j = 1,3} + i ω ρ_{o} v_{2}) e^{- i k x_{1} y_{1} / r - i k x_{3} y_{3} / r} d y_{1} d y_{3}

(47)

When the surface is rigid this result is the same as equation 15.4.15 in,¹⁴ but in this result the motion of the surface and the pumping of the fluid through the surface is included. To solve the equation, it is necessary to solve for the surface pressure on the outer surface assuming that p_s is known on the inner surface, and also to specify a model for the flow through the surface defined by the velocity v₂.

The external roughness noise

The first term in equation (47) represents the sound that is generated by surface roughness. It represents the scattering of the near field surface pressure fluctuations by the rough surface and is nonzero on both the inner and outer walls of the tunnel. The procedure for evaluating this term is given in Glegg and Devenport^12,14 and first requires the definition of the surface roughness. Since the roughness of the fabric depends on the weave pattern, which is periodic in y₁ and y₃, this may be modelled by a Fourier series expansion as

ξ (y_{1}, y_{3}) = h \sum_{m, n = - \infty}^{\infty} H_{m n} \exp (2 π i \frac{(m y_{1} + n y_{3})}{d})

(48)

where h is the mean roughness height and H_mn are a set of non-dimensional coefficients describing the roughness of the weave. Then the term in equation (47) that depends on ξ becomes

{[\frac{i k x_{j}}{r} \frac{\partial ξ}{\partial y_{j}}]}_{j = 1,3} = 2 π k h \sum_{m, n = - \infty}^{\infty} (\frac{m x_{1}}{r d} + \frac{n x_{3}}{r d}) H_{m n} e^{2 π i (m y_{1} + n y_{3}) / d}

(49)

Using this expansion in equation (47) gives the noise from the roughness as

\tilde{p} (x, ω) = \frac{- k h e^{i k r}}{r d} \sum_{m, n = - \infty}^{\infty} (\frac{m x_{1} + n x_{3}}{r}) H_{m n} \int_{- R_{1}}^{R_{1}} \int_{- R_{3}}^{R_{3}} {\tilde{p}}_{t} (y_{1}, y_{3}, ω) e^{- i κ_{1}^{(m)} y_{1} - i κ_{3}^{(n)} y_{3}} d y_{1} d y_{3}

(50)

$κ_{i}^{(m)} = k x_{1} / r - 2 π m / d$ The surface integrals then define the wavenumber spectrum of the surface pressure fluctuation, so

\tilde{p} (x, ω) = \frac{{(2 π)}^{2} k h e^{i k r}}{r d} \sum_{m, n = - \infty}^{\infty} (\frac{m x_{1} + n x_{3}}{r}) H_{m n} {\tilde{\tilde{p}}}_{t} (κ_{1}^{(m)}, κ_{3}^{(n)}, ω)

(51)

where

$\int_{- R_{1}}^{R_{1}} \int_{- R_{3}}^{R_{3}} {\tilde{p}}_{t} (y_{1}, y_{3}, ω) e^{- i k_{1} y_{1} - i k_{3} y_{3}} d y_{1} d y_{3} = {(2 π)}^{2} {\tilde{\tilde{p}}}_{t} (k_{1}, k_{3}, ω)$

is the wavenumber transform of the surface pressure. This specifies the roughness noise in terms of the wavenumber spectrum of the surface pressure fluctuations on the outer wall. However, this is driven by the surface pressure caused by the turbulent boundary layer on the inner wall, which we can model using a suitable spectrum model.¹⁴ To obtain the surface pressure on the outer wall we need to evaluate the transmission of pressure fluctuations though the wall, as was done in The Acoustic Transmission Loss. The difference in this case however is that we need to consider the net pressure fluctuations on the wall, which are generated hydrodynamically by the turbulent boundary layer.

In The Acoustic Transmission Loss we defined the pressure waves on the inside wall as the sum of the incident and reflected waves with the amplitude A+B, and calculated the transmission loss T_L=C/A to give the amplitude of the wave on the outer wall C. For roughness noise we model the surface pressure as the net pressure on the inner wall of the tunnel and relate it to the pressure on the outer wall using C=Θ(A+B). This requires the transmission coefficient Θ=C/(A+B) for a wave with a time and space dependence exp (−iωt+ik₁y₁+ik₃y₃), and this can be obtained from equation (21) as

Θ = \frac{1}{1 + K_{-}}, K_{-} = \frac{Z_{W} k_{2}^{(t)}}{ω ρ_{o}}, \frac{1}{Z_{W}} = (\frac{1}{Z} + \frac{β}{ρ_{o} c_{o}})

(52)

If the turbulence is convected subsonically with convection speed U_c then k₂^(t)=iω/U_c.

Combining equations (51) and (52) we obtain the power spectral density of the far field sound from the rough surface as

\begin{array}{l} S_{p p} (x, ω) = {(\frac{2 π k h}{r d})}^{2} \sum_{m, n = - \infty}^{\infty} {| H_{m n} |}^{2} {(\frac{m x_{1} + n x_{3}}{r})}^{2} S_{o} Φ_{p p}^{(o)} (k x_{1} / r - 2 π m / d, k x_{3} / r - 2 π n / d, ω) \\ w h e r e \\ Φ_{p p}^{(o)} (k_{1}, k_{3}, ω) = {| Θ (k_{1}, k_{3}) |}^{2} Φ_{P P} (k_{1}, k_{3}, ω) \end{array}

(53)

where Φ_pp(k₁,k₃,ω) is the wavenumber spectrum of the turbulent boundary layer surface pressure on the inner wall.

This spectrum has the same characteristics as the roughness noise from rigid surfaces described in.^13,14 First we note that it is a dipole sound source with the peak levels in the plane of the wall, and a scaling that depends on the acoustic wavenumber k. Secondly, the wavenumber spectrum is expected to be dominated by the energy at the wavenumber k₁=ω/U_c and so and if U_c<<c_o the only terms in the series summation that contribute significantly to the far field sound are those for which m=ωd/2πU_c and one expects the far field spectrum to scale on the non-dimensional frequency ωd/U_c. However, this term also has a dipole directionality that peaks along the wall and indicates that the roughness noise is zero in the direction normal to the wall.

Weave noise

In addition to roughness noise there will also be sound generated by the flow through the Kevlar fabric. This is defined by the second term in equation (47) and depends on the flow velocity through the surface, which consists of the sum of two parts. The first is the motion of the membrane which is controlled by its mechanical impedance, and the second is the flow through the membrane that is controlled by its porosity. The first of these was modeled in terms of the pressure jump across the fabric using equation (8), and the second by the flow through the weave given in equation (11), and the flow conductivity in equation (13). However, these results were given for a harmonic wave excitation and so, to be comparable to the roughness noise formulation given above, the velocity needs to be expressed in terms of the superposition of all such waves excited by the turbulent boundary layer on the inner wall. This yields a formulation based on the wavenumber spectrum of the velocity fields

{\tilde{v}}_{2} (y_{1}, y_{3}, ω) = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} {{\tilde{\tilde{u}}}_{s}^{(-)} (k_{1}, k_{3}, ω) + {\tilde{\tilde{u}}}_{p}^{(-)} (k_{1}, k_{3}, ω) f_{o} (y_{1}, y_{3})} e^{i k_{1} y_{1} + i k_{3} y_{3}} d k_{1} d k_{3}

(54)

The two velocities u_s and u_p are given by equations (8) and (13) in terms of the pressure jump across the fabric and so using equation (52) we obtain

{\tilde{\tilde{u}}}_{s}^{(-)} = \frac{(Θ - 1) {\tilde{\tilde{p}}}_{s}}{\tilde{Z}} {\tilde{\tilde{u}}}_{p}^{(-)} = \frac{(Θ - 1) β {\tilde{\tilde{p}}}_{s}}{ρ_{o} c_{o}}

(55)

Using the velocity given by equation (54) for the second term in equation (47) and evaluating the integral over the surface we obtain

\tilde{p} (x, ω) = (\frac{i ω ρ_{o} e^{i k r}}{2 π r}) ({(2 π)}^{2} {\tilde{\tilde{u}}}_{s}^{(-)} (\frac{k x_{1}}{r}, \frac{k x_{3}}{r}, ω) + \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- R_{1}}^{R_{1}} \int_{- R_{3}}^{R_{3}} {\tilde{\tilde{u}}}_{p}^{(-)} (k_{1}, k_{3}, ω) f_{o} (y_{1}, y_{3}) e^{i (k_{1} - k x_{1} / r) y_{1} + (k_{3} - k x_{3} / r) y_{3}} d k_{1} d k_{3} d y_{1} d y_{3})

(56)

From this we can calculate the sound power spectrum in the far field in terms of the surface pressure wavenumber spectrum on the inner surface, and the wavenumber spectrum of the weave porosity, f_o, as we did for roughness noise. However, the first term that depends on the motion of the surface does not couple with the acoustic far field because there is very little energy in the boundary layer pressure fluctuations at the sonic wavenumbers kx₁/r and kx₃/r. Therefore, the far field noise only depends on the flow through the membrane. The pressure on the surface can be modeled as was done for the roughness noise analysis given above and so, using equation (55) we obtain

\tilde{p} (x, ω) = (\frac{i ω e^{i k r}}{2 π r c_{o}}) \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- R_{1}}^{R_{1}} \int_{- R_{3}}^{R_{3}} β (Θ - 1) \tilde{\tilde{p_{s}}} (k_{1}, k_{3}, ω) f_{o} (y_{1}, y_{3}) e^{i (k_{1} - k x_{1} / r) y_{1} + i (k_{3} - k x_{3} / r) y_{3}} d k_{1} d k_{3} d y_{1} d y_{3})

(57)

To simplify the evaluation of these multiple integrals we can evaluate the wavenumber integrals, assuming that β and Θ are only a function of frequency, and so this equation becomes

\tilde{p} (x, ω) = (\frac{i ω}{2 π r c_{o}}) β (Θ - 1) \frac{1}{2 π} \int_{- T}^{T} \int_{- R_{1}}^{R_{1}} \int_{- R_{3}}^{R_{3}} p_{s} (y_{1}, y_{3}, t - \frac{r_{o}}{c_{o}}), f_{o} (y_{1}, y_{3}) e^{i ω t} d t d y_{1} d y_{3}

(58)

where

r_{o} = r - x_{1} y_{1} / r - x_{3} y_{3} / r

. Further insight can be obtained by considering an observer directly above the surface so that the propagation distance r_o is the same for all locations on the surface. Then using the definitions of the pressure spectra

S_{p p} (x, ω) = (\frac{π}{T}) E [| \tilde{p} |^{2}]

(59)

And the surface pressure correlation function

R_{p p} (r_{1}, r_{3}, τ) = E [p_{s} (y_{1} + r_{1}, y_{3} + r_{3}, t + τ) p_{s} (y_{1}, y_{3}, t)]

(60)

We can evaluate the far field noise spectrum by making use of the homogeneity of the turbulence over the surface. Some care however is needed to define the porosity function f_o. It cannot be assumed that the weave porosity is a perfectly distributed, it must include both a variation in effective open area for each pore and may not be regularly spaced. To allow for this we define f_o as a stochastic random variable and define a non-dimensional correlation function of the weave porosity as

R_{f f} (r_{1}, r_{3}) = \frac{1}{S_{o}} \int_{- R_{1}}^{R_{1}} \int_{- R_{3}}^{R_{3}} E [f_{o} (y_{1} + r_{1}, y_{3} + r_{3}) f_{o} (y_{1}, y_{3})] d y_{1} d y_{3}

(61)

Where S_o is the surface area of the fabric. S_o=4R₁R₃. Using these definitions gives

S_{p p} (x, ω) = {(\frac{k}{2 π r})}^{2} S_{o} {| β (Θ - 1) |}^{2} (\frac{1}{2 π} \int_{- T}^{T} \int_{- R_{1}}^{R_{1}} \int_{- R_{3}}^{R_{3}} R_{p p} (r_{1}, r_{3}, τ) R_{f f} (r_{1}, r_{3}) e^{i ω τ} d r_{1} d r_{3} d τ)

(62)

Which is the far field sound for weave noise given in terms of the cross correlation of the pressure fluctuations on the inside wall of the wind tunnel, and the weave porosity correlation. To evaluate equation (62) we first need to define the weave porosity correlation function which is unknown and hard, if not impossible to measure, and so we will consider a simple model. We will assume that the weave is periodic in both directions and defined by a single cosinusoidal variation

f_{o} (y_{1}, y_{3}) = 1 + \cos (\frac{2 π y_{1}}{d}) \cos (\frac{2 π y_{3}}{d})

(63)

where d is the weave spacing. This is perfectly periodic and uniform over the entire surface. Hence, we find that

R_{f f} (r_{1}, r_{3}) = 1 + \frac{\cos (\frac{2 π r_{1}}{d}) \cos (\frac{2 π r_{3}}{d})}{4} = 1 + \frac{1}{8} \cos (2 π (\frac{r_{1} - r_{3}}{d})) + \frac{1}{8} \cos (2 π (\frac{r_{1} + r_{3}}{d}))

(64)

and substituting this in equation (62) then gives

S_{p p} (x, ω) = {(\frac{k}{2 π r})}^{2} S_{o} {| β (Θ - 1) |}^{2} {(2 π)}^{2} (Φ_{p p} (0,0, ω) + \frac{1}{8} (Φ_{p p} (\frac{2 π}{d}, \frac{2 π}{d}, ω) + Φ_{p p} (- \frac{2 π}{d}, \frac{2 π}{d}, ω)))

(65)

Where Φ_pp(k₁,k₃,ω) is the wavenumber spectrum of the wall surface pressure

Φ_{p p} (k_{1}, k_{3}, ω) = \frac{1}{{(2 π)}^{3}} \int_{- T}^{T} \int_{- R_{1}}^{R_{1}} \int_{- R_{3}}^{R_{3}} R_{p p} (r_{1}, r_{3}, τ) e^{i ω τ - i k_{1} r_{1} - i k_{3} r_{3}} d r_{1} d r_{3} d τ

which can be modeled empirically, using a Chase or Corcos model (see¹⁴). For the Corcos model (see equation 9.2.38 of¹⁴) the spectrum peaks in region where the wavenumber is ∼ω/U_c, and is given by

Φ_{p p} (k_{1}, k_{3}, ω) = \frac{S_{p p}^{(s)} (ω) U_{c}^{2}}{π^{2} ω^{2}} \frac{α_{1} α_{3}}{[α_{1}^{2} + {(\frac{U_{c} k_{1}}{ω} - 1)}^{2}] [α_{3}^{2} + \frac{U_{c}^{2} k_{3}^{2}}{ω^{2}}]}

(66)

where the constants are

α_{1} = 0.1

and

α_{3} = 0.77

, and S_pp^(s) (ω) is the surface pressure spectrum for turbulent boundary layer fluctuations. However, the Corcos model is known to be inaccurate at very low wavenumbers and other modeling approaches suggest that the wavenumber spectrum is zero when the wavenumbers are zero. Consequently we set Φ_pp(0,0,ω)=0.

In conclusion we have derived a prediction model for weave noise given by equation (64) that depends on the wavenumber spectrum of the surface pressure fluctuations in the turbulent boundary layer on the inside surface of the wind tunnel wall. The result depends on the porosity parameter, the weave dimensions and the surface area of the window. In the following section we will compare this result to weave noise measurements.

Comparison with experimental measurements

The experimental arrangement

Experiments were conducted at the Anechoic Wall Jet Wind Tunnel (AWJT) facility of Virginia Tech¹⁵ to quantify the weave noise of a commonly used Kevlar type K120 fabric whose thread density was 34x34 TPI, open area ratio was 4% and mass per unit area was 60 g/m². A custom-made rig (see Figure 8) was built which consisted of a fabric tensioner and an aluminum frame that ensured a smooth transition from the aluminum surface to the Kevlar fabric. The area of the fabric exposed to the flow was 380 × 380 mm. A B&K type 4938 1/4-in microphone was positioned below the center of the tensioned fabric with a separation distance of 200 mm. Due to the high-frequency nature of the fabric self-noise, the microphone was in the acoustic far field, but not the geometric far field. Weave noise was observed in the results when the jet speed in the tunnel was set to 50, 60, and 70 m/s as shown in Figure 1. For a more detailed description of the experiments, we refer to Szőke et al.⁶

The surface pressure fluctuations were measured immediately upstream of the Kevlar fabric using an 1/8-in type 4138 B&K microphone flush mounted to the wall. This transducer was used under a pinhole configuration. The Helmholtz effect of the cavity below the pinhole was compensated during post-processing of the data.¹⁶ All acoustic data was collected using a B&K LAN-XI card for a time span of 32 s at a sampling rate of 131,072 Samples/s. During the post-processing of the data, the frequency resolution was set to 64 Hz using 50% of block overlapping and Hanning windowing during the Fourier transform calculations.

The properties of the flow in the AWJT at any given location can be determined using the calibration data presented in Kleinfelter et al¹⁵, which relies on the self-similarity of the wall jet flow developing over the wall. Therefore, the boundary layer properties over the fabric (boundary layer thickness, maximum velocity, friction velocity, etc.) were determined using the calibration curves provided in Kleinfelter et al.¹⁵

Comparison with the prediction model

To illustrate the effectiveness of the prediction model given by equation (64) we plot the normalized spectrum S_pp( x ,ω)r²/S_pp^(s) (ω)S_o in Figure 9, Figure 10 and Figure 11 for three different flow speeds and compare the model to the experimental results. Three different wall jet flow speeds are shown in these figures and the associated maximum speed in the wall jet, and friction velocity at the surface are given in Table 1.

Table 1.

The flow speeds for the results shown in Figure 9, Figure 10, and Figure 11.

Wall jet Speed U_jet	Max speed at Sample U_m	Friction Velocity u_τ
50 m/s	15.9 m/s	0.833 m/s
60 m/s	19.4 m/s	0.99 m/s
70 m/s	22.8 m/s	1.154 m/s

The flow speeds were determined using semi-empirical data. Wyganski¹⁹ showed that wall jet boundary layer properties can be described using power law curves. Kleinfelter²⁰ performed these curve fits for the wall jet tunnel used in these experiments. The skin friction coefficient fit was taken from Bradshaw and Gee,²¹ which then provides the friction velocity at the wall.

The convection velocity was taken to be U_c =20u_τ and β was modeled using equation (16) using $α_{W} = 0.1$ and the maximum wall jet speed at the sample, U_m.

The results clearly show that the spectral shape of the predicted weave noise matches with the experimental measurements apart from an over prediction at the peak frequency in the spectrum. This is consistent with measurements of wavenumber spectra on wavy walls ¹⁷ that showed a similar discrepancy at the convective wavenumber by measuring the roughness noise from a surface with similar scales of roughness. The errors at the peak frequency of the predicted spectrum are most probably a limitation of the Corcos model, as distinct from an error in equation (64). The levels are well predicted, but there is also some uncertainty in this result. The measurements were made in the geometric near field of the Kevlar sample and so an amplitude correction is needed to allow for the differences to the far field prediction. Similarly, the model for the grazing flow speed correction $α_{W}$ is facility dependent and so this can also impact the spectral level. The combined error of these two effects appear to cancel for the predictions shown, but their magnitude is uncertain. However, we can conclude that β is independent of frequency as suggested by equation (34).

Conclusion

In the previous sections we have addressed two important issues regarding the acoustic performance of Kevlar walled wind tunnels. The first was to address the acoustic transmission loss through the wall, when a grazing flow was present or not as the case may be. It was shown that this could be classified in terms of a non-dimensional porosity coefficient β that could be modeled in different ways depending on the experimental configuration. In the absence of grazing flow this parameter was found to depend on the open area ratio of the fabric, and, with the appropriate scaling it was found to closely match experimental results. However, in the presence of a grazing flow the same model could not be used as there will be a bias flow through the pores in the fabric. A suitable model for this was obtained in equation (34) which gives the porosity parameter as a function of the wind tunnel flow speed. While predictions based on this model were not exact, they were found to be with a few dB of the measured results over a range of flow speeds.

The second problem that is of importance in Kevlar walled wind tunnel measurements is the surface generated noise, which can be a limiting factor for aeroacoustic testing when the tunnel speed is high. There are two possible mechanisms for weave noise, roughness noise caused by the wall turbulent boundary layer pressure fluctuations being scatted into acoustic waves, and weave noise caused by the pumping of the fluid through the pores in the material. Models for both these mechanisms were given in Section 3 and were found to be dependent on the transmission of pressure fluctuations through the wall, and so also strongly affected by the porosity parameter β. Roughness noise is dipole in nature and is beamed along the wall, with a null or zero normal to the wall. The majority of wall self noise is therefore expected to be caused by the pumping mechanism. The model developed for this showed how this source could be specified using the turbulent boundary layer surface pressure wavenumber spectrum, at the wavenumber corresponding to the weave spacing. This gives a peak level at a frequency U_c/d where U_c is the convection velocity of the turbulence in the boundary layer and d is the weave spacing, and its amplitude was determined by the porosity parameter. Figures 9 to 11 show the comparison of the model for weave noise with measurements in a wall jet flow as described by Szoke et al.⁶ Good agreement is found across the frequency range, apart from at the peak frequency in the spectrum where there is an error of less than 3dB (aside from the spectral peak). However, this prediction is based on the Corcos model for the wavenumber spectrum that is known to be less accurate than other more sophisticated models, such as the Chase model, and so the error may be caused by the turbulence model that was used.¹⁷ However, from the perspective of improving the acoustic performance of the fabric used in wind tunnel walls this error is of secondary importance. The most important issue is the correct modeling of the porosity parameter and secondly choosing the weave spacing so that the peak frequency in the spectrum lies outside the operational range of the wind tunnel. There are some tradeoffs to be made here. For minimal transmission loss through the wall the value of β needs to be much greater than one, but large values of β lead to high levels of weave noise. The porosity parameter increases with the open area ratio of the fabric, but this also implies a large weave spacing that lowers the peak frequency of the weave noise. The choice of porosity, weave spacing and open area ratio, must therefore be chosen carefully to match the requirements of a particular facility.

Dedication

Shôn Ffowcs Williams had a very substantial hand in creating the field of aeroacoustics as we know it today and it is an honor to be part of this memorial issue. Two of us (SG and WD) had the privilege of interacting with Shôn in the early parts of our careers in the UK, and have vivid memories of him as a true intellectual force - a larger-than-life technical leader with a laser-sharp mind propelled by almost formidable positive enthusiasm. While this article is built on some of Shôn’s results [ref 6.] (it is hard to write much in aeroacoustics that does not), we feel that here and elsewhere we have gained much greater benefit through his inspiration and towering example.

Footnotes

Acknowledgments

The authors would like to thank the National Science Foundation, in particular Dr Ron Joslin, for their support of this research under grant CBET-2012443. In addition, we would like to thank Nandita Hari and Dr William N. Alexander for many useful discussions on this topic and assistance with the experimental measurements.

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 disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Science Foundation (CBET-2012443).

ORCID iD

Stewart Glegg

Appendix

References

Devenport

Burdisso

Borgoltz

, et al. The Kevlar-walled anechoic wind tunnel. J Sound Vibration 2013; 332(17): 3971–3991.

Brown

Devenport

Borgoltz

. Aircraft noise generation and assessment: exploitation of hybrid anechoic wind tunnels for aeroacoustic and aerodynamic measurements. CEAS Aeronaut J 2019; 10(1): 251–266.

Bahr

Hutcheson

Stead

. Unsteady propagation and mean corrections in open-jet and kevlar® wind tunnels. AIAA J 2021; 59: 1–12.

Alexander

Devenport

. Noise produced by fabric and wire mesh covered panels in low-speed anechoic wind tunnels. In: Aviation. Dallas, TX, 2015.

Nelson

. Noise generated by flow over perforate surfaces. J Sound Vibration 1982; 83(1): 11–26.

Szőke

Devenport

Hari

, et al. Investigating the Aeroacoustic Properties of Kevlar Fabrics. AIAA Aviation 2021 Forum, 2021. AIAA-2021-2255.

Ffowcs-Williams

. The acoustics of turbulence near sound-absorbent liners. J Fluid Mech 1972; 51(3): 737–749.

Howe

. Acoustics of Fluid-Structure Interactions. Cambridge: Cambridge University Press, 1998.

Jaworski

Peake

. Aerodynamic noise from a poroelastic edge with implications for the silent flight of owls. J Fluid Mech 2013; 723: 456–479.

10.

Hajian

Jaworski

. Porous panels with fixed ends lose stability by divergence. J Fluids Structures 2020; 93: 102823.

11.

Morse

Ingard

. Theoretical acoustics. Princeton University Press, 1986.

12.

Yazdchi

Srivastava

Luding

, Microstructural effects on the permeability of periodic fibrous porous media, Int J Multiphase Flow 2011; 37(8): 956–966.

13.

Glegg

Devenport

. The far-field sound from rough-wall boundary layers. Proc R Soc A: Math Phys Eng Sci 2009; 465(2106): 1717–1734.

14.

Glegg

SAL

Devenport

. Aeroacoustics of Low Mach Number Flows : Fundamentals, Analysis, and Measurement. London: Academic Press, 2017.

15.

Kleinfelter

Repasky

Hari

, et al. Development and calibration of a new anechoic wall jet wind tunnel. AIAA Scitech 2019 Forum, 2019. AIAA-2019-1936.

16.

Fritsch

Vishwanathan

Lowe

, et al. The effect of grazing flow on pinhole condenser microphones. In: AIAA Scitech 2021 Forum, 2021. AIAA-2021-0130.

17.

Devenport

Wahl

Glegg

, et al. Measuring surface pressure with far field acoustics. J Sound Vibration 2010; 329: 3958–3971. DOI: 10.1016/j.jsv.2010.03.012

18.

Brown

. Understanding and exploiting wind tunnels with porous flexible walls for aerodynamic measurement. Virginia Tech: Ph.D. thesis, 2016.

19.

Wygnanski

Katz

Horev

. On the applicability of various scaling laws to the turbulent wall jet. J Fluid Mech 1992; 234: 660–690.

20.

Kleinfelter

Russell

Hari

, et al. Development and calibration of a new anechoic wall jet wind tunnel. In: AIAA Scitech 2019 Forum, 2019. San Diego, CA. AIAA-2019-1936.

21.

Bradshaw

Gee

. Turbulent wall jets with and without an external stream, Aeronautical Research Council R&M, 1960.