Broadband noise suppression for turbofan inlet applications

Abstract

Broadband noise in turbofan inlets is represented in a frequency domain analysis by a statistical distribution of propagating duct modes. The required number of modes is estimated by Rice’s approximation method. A brief overview of essential results from the theory of acoustic propagation in ducts is given, providing a starting point for Rice’s estimation of the distribution of propagating modes in a uniform unlined duct required to represent a broadband noise source. The relationship between mode cut-off ratio and attenuation in lined ducts is examined to connect cut-off ratio to modal attenuation, and it is shown by example that for arbitrary impedance at a specified frequency, modes with similar cut-off ratios have similar attenuation. It is then shown by example that the attenuation of a low-order circumferential mode, specifically the axially symmetric circumferential mode, with a large range of propagating radial mode cut-off ratios, is insensitive to impedance variations near the optimum impedance. Two models for estimating achievable attenuation for broadband noise attenuation, based on equipartition of acoustic power among propagating modes, are considered. One uses only the axially symmetric circumferential mode, and the second uses all propagating circumferential modes and their associated propagating radial modes. A new parallel architecture version of an established propagation code, designed to include all propagating modes on a statistical basis, is described. Examples for prediction of acoustic power attenuation of broadband noise at low frequency and at moderate frequency are used to compare the performance of the axially symmetric circumferential mode and the all circumferential mode models. It is shown by example that a broadband noise model based on the axially symmetric circumferential mode predicts lower attenuation than a model with all propagating circumferential modes. Rice’s correlation of mode cut-off ratio with attenuation justifies this observation.

Keywords

Acoustics Noise Linings Inlets Attenuation

Introduction

The prediction of suppression of fan-generated noise from turbofan engines has generally been directed toward tones. For early, low bypass ratio engines, blade/stator interaction tones at approach and cut-back conditions were most important. In modern high bypass ratio engines, blade passage frequency (BPF) and multiple pure-tone noise at takeoff, cut-back and cruise conditions, as well as blade/stator interaction tones, are targeted. Performance prediction of acoustic linings has generally focused on tones, principally because the generation mechanism is well understood and the modeling of the physics of the problem lends itself to simplifications based on the Tyler and Sofrin¹ identification of the spinning mode structure associated with rotating and stationary blade counts. There has been considerable success in reduction of turbofan engine tonal noise, to the extent that it can often be attenuated to near or below the level of broadband noise. This exposes broadband noise as a contributor to the various metrics used to assess the environmental impact of aircraft noise. Consequently, it is appropriate that a prediction tool be available to assess the suppression of broadband noise that can be achieved by acoustic treatment specifically designed for it, or that can be achieved as a by-product of acoustic treatment designed for tones.

ANOPP (Aircraft Noise Prediction Program),² an aircraft noise environmental impact assessment tool, has imbedded in it an engine noise module that includes broadband noise based largely on empiricism. A user-provided engine noise module can also be used to provide data in the ANOPP format. One motivation for the present investigation has been the development of a user-based module more theoretically based than the imbedded ANOPP module.

For a specified flight operation, ANOPP predicts effective perceived noise level. This metric is based on one-third octave band spectral decomposition of engine and airframe noise. Propagation and radiation of inlet tonal contributions to the spectrum can currently be modeled efficiently by codes described in the literature^3,4 or by commercial codes. The same cannot be said for broadband inlet noise. Generation of broadband noise, and hence its propagation and radiation, is not easily quantifiable because the Tyler and Sofrin procedure for identification of modal content is not available. An early study of broadband noise,⁵ based on statistical considerations, suggested that at least at high frequencies, the acoustic field can be constructed on the basis of equipartition of acoustic power among all propagating modes. This leaves modal phasing to be treated as statistical and essentially requires representation of the source in terms of all propagating circumferential and radial modes, not just the ones associated with the Tyler and Sofrin model. The implication that models of broadband noise should include all propagating modes on a statistical basis makes codes designed for tonal noise impractical to use. The purpose of this investigation is to develop an improved method of modeling broadband noise that is useful at the design level.

The discussion begins with the assumption that it is necessary to examine inlet noise suppression over the 24 one-third octave bands with center frequencies from 50 to 10,000 Hz. Noise is generally a combination of broadband noise and tones. Tones are related to shaft frequency $\frac{Ω}{60}$ (Hz) and harmonics, including BPF, $\frac{Ω}{60} N_{B}$ and harmonics, where Ω is the shaft speed in RPM, and N_B is the number of blades. Noise generation mechanisms for tones can be identified, though the spectrum levels may be known only from measurements. For example, it is known that tones at multiples of shaft rotational speed, including BPF, can be attributed to shock structure near a supersonic tip speed fan.⁶ For these multiple pure tones, the circumferential mode orders, m, correspond to the multiplier of shaft speed. That is, the $m th$ multiple of shaft speed is generated in the $m th$ circumferential mode. These are sub-harmonics of BPF. A second predictable source of tones is related to the interaction between rotating fan blades and stationary exit guide vanes. The Tyler–Sofrin theory¹ predicts tones at frequencies that are integer multiples n of BPF, $n \frac{Ω}{60} N_{B}$ , in circumferential mode orders $m = ({nN}_{B} \pm μ N_{V})$ , μ an integer. The important point here is that for discrete tones the frequency and circumferential mode are predictable, and this is required to invoke the modeling methods conventionally used for analysis. Broadband noise is defined as spectral content that cannot be identified as tonal. For broadband noise only the frequency can be specified, circumferential mode structure is uncertain, and it generally would be assumed that all propagating circumferential modes, with their associated radial modes, are present at unknown amplitudes and phases.

The most practical design tools for modeling acoustic propagation and radiation from turbofan engines are based on frequency domain models that have a common basis in the linearized Euler equations (LEE). With the assumption of isentropic potential mean flow, LEE is reduced to an acoustic velocity potential formulation. With the further assumptions that the nacelle geometry is axially symmetric with circular or annular cross section, that the acoustic treatment is axially symmetric, and that the source is modeled in terms of circumferential modes, the field equations for the single-field variable velocity potential become spatially two dimensional for each circumferential source mode. If the duct is uniform, the mean flow field becomes uniform, and the governing field equation is the convected wave equation and can be written in terms of either acoustic potential or acoustic pressure. In the frequency domain, the formulation is completed with appropriate boundary conditions, including the specification of an impedance relationship for acoustically treated walls and the condition of vanishing acoustic particle velocity normal to untreated, rigid walls.⁷ If the potential mean flow assumption is not invoked, then in general LEE must remain in terms of the primitive flow variables. For a specified mean flow profile that does not vary axially in a uniform duct, LEE can be reduced to the Pridmore-Brown formulation for sheared flow, resulting in a single-field equation in terms of acoustic pressure.⁸ In this investigation, the sheared flow option is not used because the design tools are based on potential mean flow.

Models for acoustic propagation and radiation from turbofan inlets break down into frequency domain models and time domain models. Gabard⁹ has noted that frequency domain models are burdened by the requirement to consider all propagating circumferential modes with their associated propagating radial modes, resulting in inefficiency and computational expense. Gabard proposes the use of time domain computation and has introduced a statistical source model for this purpose. Time domain models have the advantage that the solution is a time series and Fourier methods can be used to extract spectral information, eliminating the requirement of frequency by frequency computations. Frequency domain methods have the advantage that the mathematics naturally decomposes into modal component solutions that contain a wealth of information contributing to the optimization of acoustic treatment via Rice’s classical concept of modal cut-off ratio.^10–17 It is the goal of this investigation to extend frequency domain methods to efficiently include broadband noise modeling.

Commonly used duct propagation analysis tools are based on time harmonic solutions for a specified circumferential mode. For unlined uniform ducts, each circumferential mode is associated with a distribution of radial modes. Each mode is classified as propagating (cut-on) or non-propagating (cut-off) based on whether in isolation it can carry acoustic power or not. The number of propagating radial modes associated with a specified circumferential mode decreases in number for increasing circumferential mode order, up to a circumferential mode order for which no radial modes propagate. A circumferential mode is said to propagate if at least one of its associated radial modes propagates. If no associated radial modes propagate (all modes are cut-off), the circumferential mode is cut-off. A broadband noise source in an unlined duct can be represented by all propagating modes at a specified frequency. The number of propagating modes at a specific frequency is predictable and finite, though it may be large. For a hard wall circular duct with uniform flow, Rice¹⁰ has provided a method to obtain a good estimate of the total number of modes that can propagate at a specified frequency, as well as the number of propagating circumferential modes. This estimation method and a series of Rice’s investigations of the details of the effect of mode cut-off on duct propagation and radiation^11–15 are used here to construct a useful model for broadband noise propagation and suppression.

The discussion begins with a brief overview of essential results from the theory of acoustic propagation in ducts. This provides the starting point for the method of Rice¹⁰ for the estimation of the distribution of propagating modes in a uniform unlined duct required to represent a broadband noise source. The relationship between mode cut-off ratio and attenuation in lined ducts is then examined to connect cut-off ratio to modal attenuation, and it is shown by example that for arbitrary impedance at a specified frequency, modes with similar cut-off ratios have similar attenuation. It is then shown by example that the attenuation of a low-order circumferential mode, specifically the axially symmetric circumferential mode, with a large range of propagating radial mode cut-off ratios, is quite insensitive to impedance variations. Implications of these observations for attenuation of broadband noise attenuation are discussed. With this background, two models for estimating achievable attenuation for broadband noise attenuation are discussed. Both approaches use a new parallel-processor version of an established propagation code to include all propagating modes on a statistical basis.

Propagation models

Non-uniform ducts

Models of acoustic propagation and radiation from turbofan engines that are most useful for design are based on perturbations on a steady mean potential flow. The reduction of the LEE to a two-dimensional form for axially symmetric geometry and potential mean flow is emphasized because it forms the basis for frequently used prediction codes available in the literature^3,4 and in commercial codes. The linearized conservation of mass and momentum equations, the equation of state for an isentropic medium and the acoustic velocity potential definition in terms of non-dimensional variables are

\frac{\partial ρ}{\partial t} + \nabla \cdot (ρ_{r} \vec{v}) + \nabla \cdot (ρ {\vec{M}}_{r}) = 0

(1)

p = - ρ_{r} [\frac{\partial φ}{\partial t} + {\vec{M}}_{r} \cdot \vec{v}]

(2)

ρ = - \frac{ρ_{r}}{c_{r}^{2}} [\frac{\partial φ}{\partial t} + {\vec{M}}_{r} \cdot \vec{v}]

(3)

\vec{v} = \nabla φ

(4)

The coordinate system is cylindrical, with x-axis the axis of symmetry, r the radial coordinate and θ the circumferential coordinate. Spatial coordinates $x, r$ are non-dimensional based on the reference radius R. Acoustic pressure, density, velocity and velocity potential $p, ρ, \vec{v}, φ$ are non-dimensional based on the reference radius R at the source plane and mean flow conditions $ρ_{0}, c_{0}$ at the source plane. Time is made non-dimensional using the reference length and the reference speed of sound. Local non-dimensional mean flow conditions of density $ρ_{r}$ and speed of sound c_r are based on source plane density and speed of sound, and the local non-dimensional mean flow velocity ${\vec{M}}_{r}$ is based on the source plane speed of sound. The axial component of mean flow is positive if directed in the positive x direction.

Equations (1) to (4) are supplemented by boundary conditions on the duct walls. At acoustically hard walls the normal component of acoustic particle velocity vanishes

\vec{v} \cdot \vec{n} = 0

(5)

At walls with acoustic treatment, the Myers¹⁸ impedance boundary condition is complicated by the velocity potential formulation and is presented here in three steps. The basic impedance relation is (impedance is a rigorous concept only for frequency domain models)

p = Z \frac{\partial ζ}{\partial t}

(6)

where ζ is the non-dimensional wall displacement into the wall, and the non-dimensional impedance Z is related to the dimensional impedance by

Z * = ρ_{0} c_{0} Z

. The kinematic condition derived by Myers, that enforces continuity of particle displacement, is

\vec{v} \cdot \vec{n} = \frac{\partial ζ}{\partial t} + {\vec{M}}_{r} \cdot \nabla ζ - ζ \vec{n} \cdot (\vec{n} \cdot \nabla) {\vec{M}}_{r}

(7)

where

\vec{n}

is the normal consistent with positive wall displacement. The boundary condition is completed by the acoustic momentum equation, equation (6), to relate pressure to velocity potential. Eversman¹⁹ has shown how to implement this in the finite element formulation.

Equations (1) to (7) are solved in regions where the duct and mean flow field are non-uniform, using the finite element method (FEM) in the frequency domain for a specific circumferential mode. This generates acoustic solutions of the form

φ (x, r, θ, t) = \bar{ϕ} (x, r) e i η_{r} t e - im θ

(8)

The non-dimensional frequency $η_{r} = \frac{ω R}{c_{0}}$ . x is the non-dimensional axial coordinate, r is the non-dimensional radial coordinate and θ is the angular coordinate in a cylindrical coordinate system.

FEM is implemented to generate solutions $\bar{ϕ} (x, r)$ for a single frequency and circumferential mode defined by the source. In the case of a discrete tone, the frequency and circumferential mode number are known, and the formulation is well suited for the problem. If the source is broadband it is necessary to obtain solutions over a range of frequencies, for example, one-third octave band center frequencies. In addition, no discrete circumferential mode description exists for the source, and it is supposed that all propagating circumferential modes can be present at each frequency.

Equations (1) to (8) are required in regions, where the duct and mean flow are non-uniform. The source is imposed in an appended uniform section of duct in which the flow field is essentially uniform and in which a solution in terms of a single circumferential mode and corresponding radial acoustic modes is generated, with incident and reflected modes. Incident modal amplitudes are imposed, and reflected modal amplitudes are calculated. In the case of an assumed reflection-free termination, a similar appended uniform section, perhaps with a radius differing from the source radius, is used in which the modal solution is imposed with no reflected modes and transmitted modal amplitudes calculated. For far-field radiation solutions, the computational domain is extended beyond the end of the duct with a reflection-free boundary imposed via mapped infinite elements.

The uniform duct solution yielding acoustic modes is an essential part of the general formulation of equations (1) to (8). It is also the basis for fundamental studies of the design of linings for optimum attenuation and is the starting point for the many contributions of Rice.^10–15

Propagation in a uniform duct with uniform flow

When the duct and mean flow are uniform, equations (1) to (7) lead to the convected wave equation

\frac{1}{c_{r}^{2}} [\frac{\partial}{\partial t} + {\vec{M}}_{r} \cdot \nabla] 2 φ = \nabla 2 φ

(9)

and for harmonic excitation, the convected Helmholtz equation

(1 - M_{l}^{2}) \frac{\partial 2 φ}{\partial x 2} + \nabla_{⊥}^{2} φ - 2 i M_{l} η_{l} \frac{\partial φ}{\partial x} + η_{l}^{2} φ = 0

(10)

The local Mach number M_l is based on the local speed of sound, and the local non-dimensional frequency $η_{l} = \frac{ω R_{l}}{c_{l}}$ is based on the local duct radius. Spatial dimensions are non-dimensional with respect to the local duct radius. $\nabla_{⊥}$ denotes the gradient operator for the appropriate duct cross section. This form allows the mean flow speed of sound and reference radius to be defined locally.

Equation (10) applies for circular, annular or rectangular cross section. In the following, details are given only for the circular geometry. The boundary condition at an acoustically hard wall at $r = 1$ is given by

\frac{\partial φ}{\partial r} = 0

(11)

The boundary condition for an impedance wall at $r = 1$ is

\frac{\partial φ}{\partial r} = - \frac{i η_{l}}{Z} (1 - i \frac{M_{l}}{η_{l}} \frac{\partial}{\partial x}) 2 φ

(12)

Progressive wave solutions of the form

φ (x, r, θ, t) = ϕ (r) e i η_{l} t e - {ik}_{x} x e - im θ

(13)

lead to Bessel’s equation

\frac{1}{r} \frac{\partial}{\partial r} (r \frac{\partial ϕ}{\partial r}) + [κ 2 - \frac{m 2}{r 2}] ϕ = 0

(14)

Solutions in the circular duct case are $ϕ (r) = J_{m} (κ r)$ , Bessel Functions of the first kind of order m. In the case of the hard wall duct, the transverse wave number is determined from the boundary condition, equation (11), by the transcendental equation

J_{m}' (κ) = 0

(15)

and for the duct with acoustic treatment, from equation (12)

J_{m}' (κ) = - i \frac{1}{Z} \frac{η_{l}}{κ} (1 - M_{l} \frac{k_{x}}{η_{l}}) 2 J_{m} (κ)

(16)

The non-dimensional transverse wave number κ is defined in terms of the non-dimensional axial wave number k_x by

(\frac{κ}{η_{l}}) 2 = 1 - (1 - M_{l}^{2}) (\frac{k_{x}}{η_{l}}) 2 - 2 M_{l} (\frac{k_{x}}{η_{l}})

(17)

The inverse relationship is

\frac{k_{x}}{η_{l}} = \frac{1}{1 - M_{l}^{2}} [- M_{l} \pm \sqrt{1 - (1 - M_{l}^{2}) (\frac{κ}{η_{l}}) 2}]

(18)

For a specified circumferential mode, m, equation (15) or (16) yield an infinite sequence of transverse wave numbers (eigenvalues), $κ_{mn}$ , eigenfunctions, $ϕ_{mn} = J_{m} (κ_{mn} r)$ and the associated axial wave numbers, $k_{x_{mn}}$ , completing the $m, n$ modal solution of the form of equation (13). In the case of the hard wall duct, $κ_{mn}$ are real, and for the acoustically treated duct, $κ_{mn}$ are in general complex. For the hard wall duct, the sequence of eigenvalues $κ_{mn}$ for a specified circumferential mode m increases in magnitude with the radial mode order n. For increasing circumferential mode order m, the sequence shifts to larger magnitudes for corresponding radial mode order n. The rule of thumb for the hard wall duct is that the magnitude of $κ_{mn}$ increases with mode order as classified by the combination $m, n$ .

In an approach consistent with the FEM formulation for the non-uniform duct suggested by equations (1) to (8), the eigenvalue problem is solved via an FEM formulation.²⁰ This is done in a form common to unlined and lined ducts with uniform flow. The axial wave number $k_{x_{mn}}$ is the eigenvalue, and the transverse wave number is calculated from equation (17).

The axial wave number k_x determines the type of solution in the unlined duct. Since the transverse wave number κ is real, equation (18) reveals that if $(1 - M_{l}^{2}) (\frac{κ}{η_{l}}) 2 > 1$ the axial wave number k_x becomes complex, and the assumed progressive waves of equation (13) decay in the $\pm x$ direction. In fact, further analysis shows that if this is the case, no acoustic power is carried in an isolated mode. The mode is cut-off and deemed non-propagating. The critical value of $κ_{c}$ above which the mode is non-propagating is

κ_{c} = \frac{η_{l}}{\sqrt{(1 - M_{l}^{2})}}

(19)

The ratio $κ_{c} / κ$ is defined as the cut-off ratio ξ and is

ξ = \frac{η_{l}}{κ \sqrt{(1 - M_{l}^{2})}}

(20)

where

ξ > 1

(

κ < κ_{c}

) corresponds to propagation, and

ξ < 1

(

κ > κ_{c}

) corresponds to cut-off. The rule of thumb that follows from the previous discussion of the sequence of transverse eigenvalues

κ_{mn}

is that for the hard wall duct the magnitude of

κ_{mn}

increases with increasing mode order, and the cut-off ratio decreases. The magnitude of the critical transverse eigenvalue

κ_{c}

increases with non-dimensional frequency, and therefore for a given mode order, cut-off ratio increases with increasing non-dimensional frequency and decreases with decreasing non-dimensional frequency.

A model for broadband noise must necessarily consider many, if not all, propagating modes in a statistical distribution. Rice¹⁰ provides an estimation of the number of modes this could include.

Modal distribution in unlined ducts

There is no Tyler and Sofrin¹ model for broadband noise, and it is assumed that all propagating modes at a specific frequency must be considered. Rice¹⁰ has provided a basis for estimation of the number of modes this might be for unlined ducts. His analysis introduces a parameter $α_{0}$ , identified as the critical value of the transverse wave number, $κ_{c}$ (equation (19))

α_{0} = κ_{c} = \frac{η_{l}}{\sqrt{(1 - M_{l}^{2})}}

(21)

This definition is retained here to facilitate comparison with Rice’s work. (Comparison with Pridmore-Brown⁸ will show that the present non-dimensional frequency, $η_{l}$ , is related to Rice’s non-dimensional frequency by a factor of π.) He shows that for a defined circumferential mode m, duct Mach number M_l and non-dimensional wave number $η_{l}$ (fixing $α_{0}$ ), an estimate of the number of propagating radial modes N_m is obtained from

N_{m} (\frac{m}{α_{0}}) = \frac{α_{0}}{π} (\sin (\cos - 1 (\frac{m}{α_{0}})) - \frac{m}{α_{0}} \cos - 1 (\frac{m}{α_{0}})) + \frac{1}{4}

(22)

The eigenvalues of equation (15) are insensitive to the sign of m, corresponding to progressive wave solutions of equation (13) that include the axially symmetric mode for $m = 0$ and positive and negative spinning modes for $m \neq 0$ . Equation (22) applies for $m \geq 0$ , that is for the axially symmetric mode and positive spinning modes. The count of negative spinning modes is the same as for positive spinning modes. Equation (22) is therefore used to count the total number of propagating radial modes for a specified circumferential mode m, including the axially symmetric mode $m = 0$ and the positive spinning modes $m > 0$ . By the extension above the number of negative spinning modes is determined for $m < 0 .$ In the following, m, considered as positive, is referred to as the circumferential mode number with the understanding that for $m \neq 0$ ; it implies the combination of positive and negative spinning modes.

Equation (22) can be solved by various means to determine the circumferential mode number for which there is only one propagating radial mode by setting $N_{m} (\frac{m}{α_{0}}) = 1$ . This estimates the highest order propagating circumferential mode $m_{max}$ for a specified $α_{0}$ . The number of propagating circumferential modes is $M_{c} = m_{max} + 1$ . Following the discussion above, the total number of propagating circumferential modes, including positive and negative spinning modes, is $2 m_{max} + 1$ . In a simple graphical approach, equation (22) is shown for a useful range of $α_{0}$ in Figure 1. Rice notes that the approximation is good down to $α_{0} = 10$ and is best at high frequencies. $α_{0} = 65$ is in the range of three times BPF for most engines. The figure in this form is particularly useful. It can be used to estimate the number of propagating radial modes for a specific circumferential mode. For a specified $α_{0}$ , the corresponding curve provides the number of propagating radial modes as a function of $(m / α_{0})$ The y-axis intercepts $N_{m} (0)$ represent the number of radial modes for the axially symmetric circumferential mode, $m = 0$ . When $N_{m} (\frac{m}{α_{0}}) < 1$ , no modes propagate. For a specified $α_{0}$ , the value of $\frac{m}{α_{0}}$ for which a curve intersects the line $N_{m} (\frac{m}{α_{0}}) = 1$ provides an estimate of the highest order circumferential mode $m_{max}$ that has at least one propagating radial mode. In other words, the highest order propagating circumferential mode. For example, for the highest frequency $α_{0} = 65$ , Figure 1 can be used to approximate the highest order propagating circumferential mode $m_{max}$ . This is done by finding the intersection of the $α_{0} = 65$ curve with $N_{m} (\frac{m}{α_{0}}) = 1$ at approximately $\frac{m}{α_{0}} = 0.88$ . This results in the estimate $m_{max} \approx 0.88 α_{0} = 57$ , subject to the accuracy of estimating the intercept.

A useful point to be made is that for high frequency, and consequently high $α_{0}$ , this intersection moves out on the $\frac{m}{α_{0}}$ axis, suggesting that the highest order propagating circumferential mode tends toward becoming an integer $m_{max}$ numerically close to $α_{0}$ (actually closer to η for typical inlet Mach numbers). For a typical inlet Mach number of 0.5, $α_{0} = 65$ corresponds to $η = 56.3$ , very close to the estimate for $m_{max}$ deduced by reading the figure.

Rice also provides a prediction for the total number of modes, composed of the propagating circumferential modes with their propagating radial modes. The prediction formula for one direction (including the axisymmetric mode $m = 0$ ) is

\tilde{N} (α_{0}) = \frac{α_{0}^{2}}{8} + \frac{1}{2} (\frac{1}{2} + \frac{1}{π}) α_{0}

(23)

Equation (23) is shown in Figure 2. This figure shows the dramatic rise in the number of propagating modes as $α_{0}$ increases. At the largest value of $α_{0}$ shown in Figure 1, $α_{0} = 65$ , the approximate number of propagating modes from Figure 2 is 555 (computed from equation (23)).

Another useful approximation given by Rice is the number of propagating radial modes in the axially symmetric circumferential mode, $m = 0$ , obtained directly from equation (22)

N_{m} (\frac{m}{α_{0}})_{m = 0} = N_{0} (α_{0}) = \frac{α_{0}}{π}

(24)

Equation (24) shows the transition from the notation of equation (22) and retains Rice’s notation in the final form. At $α_{0} = 65$ , this yields to the nearest integer $N_{m} = 21$ , and this can be read quite accurately from Figure 1 at the intercept on the y-axis.

The approximate total number of modes predicted, including the axially symmetric mode and both clockwise and counter-clockwise spinning modes, is twice the result of equation (23), less the prediction of equation (24)

N_{T} (α_{0}) = \frac{α_{0}^{2}}{4} + \frac{α_{0}}{2}

(25)

These results illustrate the rapidly expanding computational problem encountered in developing a model of broadband noise propagation. As frequency increases, the overall number of propagating modes increases dramatically, as does the number of propagating circumferential modes. In the $α_{0} = 65$ case referred to above, the FEM propagation solution requires consideration of approximately 57 circumferential modes and the associated 555 propagating radial modes. It is clear that if a useful model of broadband noise propagation is the goal, economies in the formulation are essential.

Modal distribution and attenuation in lined ducts

Rice¹¹ has studied broadband noise attenuation based on a semi-infinite, uniform circular cross-section lined duct. In the present study, the duct configuration is extended to consist of a finite lined section imbedded between an unlined source section and an unlined, reflection-free termination section. A distribution of many propagating modes is assumed in the source section, incident on the lined section, where there is scattering into transmitted modes into the lined section and reflected modes back to the source section. At the end of the lined section, there is scattering into reflected waves in the lined section and transmitted modes in the reflection-free termination section. This configuration was chosen for several reasons. It is physically realistic, representing a short section of unlined duct between the fan source and the lining. It is compatible with available FEM analysis methods. Most importantly, in the unlined source section all incident acoustic power is accounted for in the incident cut-on modes, and the number of these can be estimated by Rice’s method.¹⁰

Rice’s estimation of the number of propagating modes is limited to unlined ducts, and therefore appropriate for the source and termination sections of the current duct model. The analysis method used in the present study models the source plane and termination plane with unlined duct modes, but uses a full FEM propagation solution in the interior, so that scattering at the lining leading and trailing edges is implicit in the solution. Modal solutions in the lined section are not explicitly generated. However, some features of a modal solution in the lined section are useful for envisioning how broadband noise is attenuated.

For lined ducts, the clear distinction between cut-on and cut-off modes, and the implication for transmission of acoustic power, becomes blurred. Since the transverse wave number, κ, is complex for a lined duct, the conventional definition, equation (20), no longer leads to a real cut-off ratio. Rice¹³ has extended the definition of cut-off ratio to the case of lined ducts when the transverse wave number is complex. Essentially, the approach is to use the real part of the transverse wave number in the definition of cut-off ratio. Rice has presented a large body of work correlating cut-off ratio with optimum lining impedance and far-field radiation directivity based on this extended definition.^12–17 Modal distribution estimates for unlined ducts serve only as a guide for lined ducts.

As an example that can be easily visualized, unlined and lined ducts at relatively low frequency are examined to show the relationship between modal distributions in the two cases. The configuration is a circular duct with inlet flow at

M = 0.5

(negative Mach number in the defining equations) and non-dimensional frequency

η_{l} = 12

(α_{0} = 13.86)

. Table 1 shows the cut-off ratio for all propagating modes for the unlined duct. Columns represent the 12 propagating circumferential modes,

m = 0

m = 11

. For each propagating circumferential mode, the rows represent the cut-off ratios for propagating radial modes, numbered from

n = 1

. There are a total of 29 propagating modes. The axially symmetric mode,

m = 0

, has five propagating radial modes and

m = 8

through

m = 11

have only one propagating radial mode.

Table 1.

Cut-off ratios for propagating modes for an unlined circular duct, inlet $M = 0.5, η_{l} = 12, α_{0} = 13.86$ .

	m = 0	1	2	3	4	5	6	7	8	9	10	11
n = 1	∞	7.53	4.54	3.30	2.61	2.26	1.85	1.62	1.44	1.29	1.18	1.08
2	3.62	2.60	2.07	1.73	1.49	1.32	1.18	1.07
3	1.98	1.62	1.39	1.22	1.09
4	1.36	1.18	1.05
5	1.04

Columns are circumferential modes from $m = 0$ to $m = 9$ . Rows are cut-off ratios for radial modes from $n = 1$ .

Table 2 shows cut-off ratio and attenuation in dB per duct radius for the same duct with arbitrarily chosen impedance

Z = 2.0 - i 1.0

. Columns again represent circumferential mode number, but rows are structured so that for each radial mode number, the upper row shows cut-off ratio and the second row represents attenuation per unit non-dimensional duct length in dB. Cut-off ratio is defined according to the convention used by Rice,¹¹ although as he points out true cut-off does not occur in the sense that power transmission is blocked for cut-off ratio less than unity. Propagating modes are still identified according to cut-off ratio greater than unity.

Table 2.

Cut-off ratios and attenuation for propagating modes for a lined circular duct, inlet $M = 0.5, η_{l} = 12, α_{0} = 13.86$ , $Z = 2.0 - i 1.0$ .

	m = 0	1	2	3	4	5	6	7	8	9
n = 1	5.64	3.54	2.64	2.12	1.78	1.54	1.35	1.21	1.09	1.01
	0.19	0.50	0.95	1.60	2.52	3.89	6.07	10.08	19.41	39.28
2	2.45	1.93	1.60	1.38	1.21	1.07
	1.13	2.04	3.43	5.72	10.15	22.78
3	1.56	1.32	1.14
	3.74	6.78	13.85
4	1.12
	15.46

Columns are circumferential modes, $m$ = 0 to 9. Rows are radial modes from $n = 1$ with cut-off ratio (above) and attenuation in dB per unit non-dimensional duct length (below).

The number of propagating circumferential modes is reduced from 12 to 10, and the total number of propagating modes is reduced from 29 to 20. It is clear that attenuation is correlated with cut-off ratio across the circumferential and radial modes, with cut-off ratios near unity producing large attenuation. The insertion of acoustic treatment compresses the range of cut-off ratios from above. It is observed that attenuation increases rapidly as cut-off ratio reduces toward unity, and the highest circumferential modes have large attenuation.

The information shown in Table 2 is partially reproduced in Figure 3. Shown is the relationship between attenuation and cut-off ratio for the case of Table 2. Missing for the sake of clarity in the plot is the information relating the attenuation and cut-off ratio to specific circumferential/radial mode orders $m, n$ . This can be obtained by reference back to Table 2. One method of ordering modes by order is to use cut-off ratio, with decreasing cut-off ratio corresponding to increasing mode order. With this convention, Figure 3 provides a convincing demonstration of the correlation of attenuation with mode order and, specifically, the increase of attenuation with increasing mode order.

Details of the scattering of unlined duct source modes into the lined section are not explicitly part of the propagation solution. The cut-off ratios and attenuations serve to indicate what type of propagation the lined duct will support. It is known that a specific circumferential mode will scatter into the same circumferential mode in the lined section. It is clear that the high-order circumferential modes will have high attenuation (particularly those with just one propagating radial mode). Among the lower circumferential modes, attenuation will depend on the distribution of radial modal amplitudes, and this can be analyzed statistically.^19–21

Sensitivity to lining impedance

For broadband noise, it has been found by extensive example calculations that achievable attenuation is quite insensitive to impedance. Figure 4 provides a good example of this for a mode amplitude and phase distribution that produces the mean attenuation over a sample space of 1000 randomly chosen amplitudes and phases. This impedance map for a circular duct at non-dimensional frequency $η = 40.87$ , non-dimensional lining length $L / R = 1$ and inlet Mach number $M = 0.5$ shows contours of equal attenuation in dB as a function of lining resistance (y-axis) and reactance (x-axis), centered on the optimum impedance for the axially symmetric circumferential mode, $m = 0$ . This circumferential mode has a large number of propagating radial modes (for the unlined duct section representing the source) with a broad distribution of cut-off ratios.

For the broad range of cut-off ratios represented by this axially symmetric circumferential mode, attenuation is insensitive to impedance variations, roughly 0.1% change in attenuation for a 2.5% change in resistance or reactance near the optimum. An extended representation of broadband noise, including all propagating circumferential modes, introduces many more radial modes but does not extend the range of cut-off ratios with which attenuation is correlated. The examples of Tables 1 and 2 support this. For a given frequency, higher order circumferential modes have a reduced number of propagating radial modes, and attenuations for these circumferential modes become increasingly sensitive to impedance. These higher order circumferential modes can have attenuations typical for rotor locked tones, such as multiple pure tones, for which the circumferential mode number can approach the critical transverse wave number $α_{0}$ in magnitude. For high frequencies, these sensitive circumferential modes are only a few out of a large number of propagating circumferential modes and do not alter the observation based on just the axially symmetric circumferential mode that broadband noise attenuation is insensitive to impedance variations. The expectation is that, at least for high frequencies with many propagating modes, at a fixed frequency broadband attenuation is insensitive to variations in impedance.

Broadband noise

Rice¹¹ has suggested that broadband noise can be represented by the axially symmetric circumferential mode $m = 0$ . His computational model of a semi-infinite lined duct $0 \leq x \leq \infty$ did not focus on a statistical representation of a broadband noise source but rather on a deterministic distribution of axially symmetric radial modes at $x = 0$ with complex amplitudes determined to produce a plane wave. In the present study, the model is extended to include an unlined source section, a lined section of length L, and an unlined, non-reflecting termination section. In the unlined source section, the modal structure is constructed to statistically represent broadband noise. There are two principal differences from Rice’s model. First, the source is statistically represented, and second, scattering is included in the computational model at the leading and trailing edges of the lined section. Two levels of approximation are considered. In the first case, following the suggestion of Rice, broadband noise is assumed to be axially symmetric, with all propagating radial modes in the axially symmetric circumferential mode randomly distributed. In the second case, broadband noise consists of all propagating circumferential modes, each with propagating radial modes randomly distributed.

The two models of broadband noise are examined using a duct propagation code based on the formulations described in equations (1) to (8). The source is defined in terms of mode amplitudes and phases and is propagated through the lined section via an FEM implementation. Scattering at the lining leading and trailing edges is imbedded in the solution. In its original implementation,^21–23 the code architecture is sequential, considering one frequency and one circumferential mode at a time. In the implementation reported here, the code is parallel architecture with the number of processors determined by the number of cases of frequency and circumferential mode. To represent broadband noise, frequency is fixed, and solutions for duct propagation in all propagating circumferential modes, determined from Rice’s estimate,¹⁰ are generated and superposed on an acoustic power basis. With this approach, the inclusion of all propagating circumferential modes (each with its associated propagating radial modes) means that all propagating modes are used to represent broadband noise. The number of propagating circumferential modes determines the number of processors used.

The observation of Dyer⁵ that broadband noise can be represented by propagating modes with equal power and random phase is the basis for the statistical model used here. The source for a specified circumferential mode consists of all corresponding propagating radial modes with equal power and random phases. A phase distribution for the circumferential mode is determined by generating a sample space of 1000 randomly constructed source phase distributions and forming solutions for the acoustic field for each one. The ensemble of 1000 solutions is post-processed to find the solution with the mean attenuation over the 1000 cases. The phase distribution for this case is taken as the phase distribution defining the circumferential mode. This leaves the scaling of the acoustic power in the circumferential modes to be determined.

The propagation model is non-dimensional throughout, as noted in Propagation Models section. The statistical model is based on equipartition of non-dimensional acoustic power and random phase for propagating radial modes. Conveniently, the non-dimensional acoustic power in each radial mode is unity, so that the total non-dimensional power in each circumferential mode is equal to the number of propagating radial modes in the circumferential mode and the total non-dimensional acoustic power is equal to the total number of propagating radial modes across all propagating circumferential modes. Connecting propagation solutions to a source model requires source data. The ideal situation occurs if the source is specified by an acoustic power spectrum $W (f)$ or sound power level spectrum $SWL (f)$ . Because of equipartition of power, the source power is a scale factor S_f multiplying the total unscaled acoustic power contributed by all propagating radial modes across all propagating circumferential modes

W (f) = S_{f} (f) N_{T}

(26)

where N_T is the total number of propagating radial modes and is frequency dependent via the critical transverse wave number

α_{0}

, as noted in the developments of equations (21) to (25).

S_{f} (f)

carries the units of acoustic power, since the source data are presumably dimensional.

The field equations for propagation are independent of the sign of the circumferential mode number. The statistics of the acoustic power distribution among the radial modes are therefore independent of the sign of the circumferential mode number, and it is only necessary to generate solutions for positive values of m. N_T includes the axially symmetric circumferential mode contribution N₀, estimated by Rice according to equation (24), and the contributions of all propagating positive and negative spinning modes

N_{T} = N_{0} + 2 \sum_{m = 1}^{m_{max}} N_{m}

(27)

where

m_{max}

is the highest order propagating circumferential modes with

m > 0

. N_m is the number of propagating radial modes in the propagating circumferential mode m,

m > 0

, estimated by Rice according to equation (22).

The scale factor $S_{f} (f)$ is easily determined in terms of source acoustic power or source SWL

S_{f} (f) = \frac{W (f)}{N_{T}} = \frac{10 - 12 10 (\frac{SWL (f)}{10})}{N_{T}}

(28)

Acoustic power is distributed in the propagating circumferential modes based on the number of propagating radial modes in the circumferential mode

W_{0} = S_{f} (f) N_{0}, m = 0 W_{m} = 2 S_{f} (f) N_{m}, m > 0

(29)

where W₀ is the acoustic power distributed to the axially symmetric circumferential mode, and W_m is the acoustic power distributed to the circumferential modes

(+) m

and

(-) m

. N₀ and N_m can be estimated by equations (24) and (22); however, they are calculated exactly in the FEM propagation code reported here. Once S_f is determined on the basis of counting the propagating modes, the acoustic power in each propagating radial mode, the acoustic power in each propagating circumferential mode and the scaled acoustic field consistent with the source power are available.

It is more likely that the source is defined by a sound intensity level ( $SIL$ ) or sound pressure level spectrum at a measurement microphone (for example, on the duct wall near the fan plane). To connect with this situation, information is presumed available in the form of a $SIL$ spectrum. In this case, the idea of equal acoustic power distribution is relaxed, and the $SIL$ spectrum is taken as the superposed contribution of all propagating circumferential modes, ${SIL}_{m}$ . It is necessary to distribute the acoustic intensity at the measurement location among the propagating circumferential modes. This is initiated by scaling each circumferential mode to produce unit acoustic pressure magnitude at the measurement microphone. It is then assumed that the distribution of intensity at the measurement location among the circumferential modes is determined by the ratio of the number of propagating radial modes in the circumferential mode to the total number of propagating radial modes in all propagating circumferential modes. These ratios are weighting factors for each circumferential mode, including the axially symmetric mode and spinning modes in both directions corresponding to positive and negative values of the circumferential mode number m. For the axially symmetric mode, the weighting factor is

w_{0} = \frac{N_{0}}{N_{T}}

(30)

where N₀ and N_T appear in equation (27). For the spinning modes, the weighting factors are

w_{m +} = \frac{N_{m +}}{N_{T}} and w_{m -} = \frac{N_{m -}}{N_{T}}

(31)

corresponding to positive and negative values of the circumferential mode number m. This makes it possible for the spinning modes to define weighting factors that combine the positive and negative circumferential modes as

w_{m} = \frac{2 N_{m}}{N_{T}}, m > 0

(32)

The contribution of circumferential mode m to the $SIL$ spectrum is obtained as

{SIL}_{m} = SIL + 10 \log w_{m}

(33)

where w_m is the weighting factor from equation (30) or (32) depending on whether the circumferential mode is axially symmetric or spinning.

SIL

is the measured Sound Intensity Level, and

{SIL}_{m}

is the reference Sound Intensity Level contribution of circumferential mode m. Propagation code solutions for acoustic pressure for each circumferential mode are scaled to have this reference Sound Intensity Level at the measurement point.

As noted, this relaxes the idea of equal partition of acoustic power among all propagating modes. It retains equal partition of power among radial modes for each circumferential mode but scales the circumferential modes to be consistent with the Sound Intensity Level defining the source. The assumption is that intensity at the measurement location distributes much like acoustic power. The failing is that intensity is a point metric at the measurement location, while power is a global metric at a duct cross section. Inconsistency with the concept of equal partition of acoustic power arises for circumferential modes that have low intensity at the measurement location. This can occur because the propagation solutions depend on the statistics of the source distribution. While not retaining a strict dependence of propagating modal power on the number of propagating radial modes across all propagating circumferential modes, results for acoustic power in each circumferential mode generally show that lower order circumferential modes with many propagating radials tend to dominate the computed source SIL spectrum.

The parallel architecture code also is suitable for an axially symmetric model for broadband noise in which only a single circumferential mode is used, as proposed by Rice.¹¹ If the example with $α_{0} = 65$ , referred to earlier, is used, it is estimated that 57 circumferential modes are required to include all 555 propagating modes. If, as Rice suggests, broadband noise is assumed axially symmetric, only the circumferential mode $m = 0$ with 21 propagating radial modes is used.

The parallel architecture code for broadband noise is an evolution of a code for propagation of multiple circumferential modes at different frequencies used to model multiple tones, including the effect of lining non-linearity.²⁴ Rice’s model for broadband noise¹¹ using only the axially symmetric mode suggests that broadband noise could be included in the multiple tone analysis by adding axially symmetric modes at the one-third octave band center frequencies. The investigation here was at least partially motivated by the intent to examine the effect on predicted attenuation of broadband noise by this simplification of Rice’s axially symmetric mode approach.

A realistic acoustic lining design, previously presented in connection with tonal noise suppression,²² is used here to assess lining performance for broadband noise. The inlet duct has a uniform circular cross section. In Zlavog and Eversman,²² the duct shape was an actual, slightly non-uniform, inlet contour. In the present example, the inlet contour is taken as uniform to maintain consistency with the discussion to this point. The source radius is 27.7 in. (0.703 m), and the fan has 22 blades with shaft speed $Ω = 6000 RPM$ (100 rev/s) and fan tip rotational Mach number 1.30 based on fan face speed of sound 1117 feet per second (340.5 m/s). The fan face inlet Mach number is 0.55. Acoustic treatment is installed on the inlet inner surface over the axial distance $0.12 \leq x / R \leq 0.90$ , where R is the source plane radius at the source plane, $x = 0$ . The lining is two degree of freedom, with impedance based on an impedance model similar to the one reported by Yu et al.²⁵

The source Sound Pressure Level spectrum for this example (assumed equivalent to an SIL spectrum) is in 50 Hz bands over the range $50 \leq 2500$ Hz, shown in Figure 5. There are four dominant tones with circumferential modes $m = 9, 11, 16, 22$ at frequencies 900, 1100, 1600 and 2200 Hz; 2200 Hz is BPF. The corresponding non-dimensional frequencies are $η = \frac{2 π n \frac{Ω}{60} R}{c} = 11.7, 14.3, 20.8, 28.6$ . There are also lower level tones at 100 Hz intervals starting with 100 Hz, representing predictable multiples of shaft speed, in circumferential modes $1 \leq m \leq 25$ . All other band levels represent broadband noise.

The lining is designed to target the 1100 Hz and 2200 Hz tones. It is two degree of freedom with perforated plate face sheet and laser drilled septum. Table 3 shows the lining parameters chosen to best achieve the suppression objective for the targeted tones. Figure 6 shows the impedance spectrum for the lining design over the frequency range 50 < f < 2500 Hz.

Table 3.

Design parameters for lining designed for tones.

	Lining parameters for assumed spectrum
Face sheet fraction open area	0.035
Face sheet hole diameter, in.(cm)	0.04 (0.102)
Face sheet thickness, in.(cm)	0.04 (0.102)
BL momentum thickness, in.(cm)	0.079 (0.200)
Septum insertion depth, in.(cm)	0.25 (0.635)
Septum fraction open area	0.025
Septum hole diameter, in.(cm)	0.007 (0.018)
Septum thickness, in.(cm)	0.029 (0.074)
Septum backing depth, in.(cm)	0.95 (2.42)

Propagation of broadband noise at 450 Hz and 1650 Hz is modeled. The frequency 450 Hz corresponds to non-dimensional frequency $η_{l} = 5.84$ ( $α_{0} = 6.99$ ), and 1650 Hz corresponds to $η_{l} = 21.42$ ( $α_{0} = 25.65$ ). Reference to Figure 6 shows that at 450 Hz the lining impedance is near $Z = 5.0 - i 4.0$ , and at 1650 Hz it is near $Z = 5.0 - i 1.0$ . At 450 Hz, the value of $α_{0}$ is below the limit Rice suggests for accurately estimating the number of propagating unlined duct modes ( $α_{0} = 10$ ). The Rice estimates for the number of propagating circumferential modes $M_{c} = m_{max} + 1 = 4$ ( $m \geq 0$ ), number of propagating radial modes for circumferential mode $m = 0$ , $N_{0} = 2$ , and total number of propagating modes, $N_{T} = 14$ ( $- m_{max} \leq m \leq m_{max}$ ), compared to the actual number of propagating circumferential modes $M_{c} = 6$ , number of propagating radial modes for the axially symmetric circumferential mode, $N_{0} = 2$ and total number of propagating modes $N_{T} = 16$ , are still useful. Only the number of propagating circumferential modes M_c is required to be accurate to assure that all propagating modes are included. The propagation code keeps track of the number of propagating radial modes for each propagating circumferential mode, and N_T is available accurately.

For 1650 Hz, the Rice estimates are for the number of propagating circumferential modes $M_{c} = 21$ ( $m \geq 0$ ), number of radial modes in the axially symmetric circumferential mode $N_{0} = 8$ , and total overall number of propagating modes $N_{T} = 176$ ( $- m_{max} \leq m \leq m_{max}$ ). Execution of the broadband propagation code predicts the number of circumferential propagating modes at $M_{c} = 24$ , supports the Rice estimate of $N_{0} = 8$ , and slightly under-predicts the Rice estimate of the total number of propagating modes with $N = 178$ . The extra circumferential modes have only one radial mode, and it appears that the Rice estimate, with the necessity to round or truncate to obtain integer numbers of modes, under-predicts the number of propagating circumferential modes.

In the following examples, the actual propagating circumferential mode count was used, so that all propagating modes are present. Comparisons with computations using the Rice estimate for circumferential modes, which slightly underestimates the number, show very little difference in conclusions about achievable attenuation of acoustic power, particularly at higher frequency, where the extra unaccounted for modes are a small proportion of a large number of propagating modes.

For broadband noise in the axially symmetric circumferential mode, the parallel code is used with just the single frequency and circumferential mode. In the low frequency case,

f = 450 Hz

, the short summary of Table 4 is given. This and following tables are organized in columns showing the circumferential mode (

m = 0

only in this case), number of propagating radial modes for the corresponding circumferential mode (two in this case), the reference Sound Intensity Level (91.75 dB in this case), the source Sound Power Level (SWL In) in this circumferential mode (92.60 dB in this case), and the transmitted Sound Power Level (SWL Out) (87.00 dB in this case). The reference Sound Intensity Level is the Sound Intensity Level distributed to this mode via the weighting factors and equation (33). In this case, the single circumferential mode with two propagating radial modes accounts for the entire measured Sound Intensity Level at the measurement point.

Table 4.

Acoustic power summary for axially symmetric broadband noise, 450 Hz, $η = 5.84$ , $M = - 0.55$ .

Circ. mode	No. rad. modes	Reference SIL dB	SWL In dB	SWL Out dB
0	2	97.75	92.60	87.00
Overall	2	97.75	92.60	87.00

SWL In: source Sound Power Level; SWL Out: transmitted Sound Power Level.

SWL In for circumferential mode m (

m = 0

in this case) is the source acoustic sound power level in the circumferential mode. This is obtained by scaling the circumferential mode m acoustic pressure solution so that the Sound Intensity Level at the measurement point is equal to the reference Sound Intensity Level for the circumferential mode m. The relationship between reference

SIL

and input acoustic power in a circumferential mode depends on the statistical propagation solution. Input SWL can be higher or lower than the reference

SIL

, depending on the radial modal phase distribution in the statistical propagation solution and the resulting spatial details of the acoustic pressure field. SWL Out for the circumferential mode is computed from SWL In, and the power attenuation results resulting from the statistical propagation solution. The overall reference Sound Intensity Level (over all circumferential modes) is consistent with the defined source spectrum at this frequency. Overall input power and transmitted power is obtained by the usual superposition of circumferential modal powers. In this axially symmetric case attenuation of acoustic power is about 5.6 dB. Table 4 is particularly simple with only one circumferential mode. Table 5 with six circumferential modes is laid out in exactly the same way, as are Tables 6 and 7 for a higher frequency example.

Table 5.

Acoustic power summary for broadband noise with all propagating circumferential modes, 450 Hz, $η = 5.84$ , $M = - 0.55$ .

Circ. mode	No. rad. modes	Reference SIL dB	SWL In dB	SWL Out dB
0	2	88.72	83.57	77.97
1	4	91.73	84.38	78.74
2	4	91.73	83.23	75.19
3	2	88.72	82.04	65.20
4	2	88.72	81.08	62.62
5	2	88.72	80.34	58.39
Overall	16	97.75	90.45	82.46

SWL In: source Sound Power Level; SWL Out: transmitted Sound Power Level.

Table 6.

Acoustic power summary for axially symmetric broadband noise 1650 Hz, $η = 21.42$ , $M = - 0.55$ .

Circ. mode	No. rad. modes	Reference SIL	SWL In dB	SWL Out dB
0	8	91.75	89.77	87.72
Overall	8	91.75	89.77	87.72

SWL In: source Sound Power Level; SWL Out: transmitted Sound Power Level.

Table 7.

Acoustic power summary for broadband noise with all propagating circumferential modes, 1650 Hz, $η = 21.42$ , $M = - 0.55$ .

Circ. mode	No. rad. modes	Reference SIL	SWL In dB	SWL Out dB
0	8	78..28	76.29	74.24
1	16	81.29	85.85	83.40
2	14	80.71	87.92	85.61
3	14	80.71	78.17	75.19
4	14	80.71	72.77	69.34
5	12	80.04	76.09	72.60
6	12	80.04	74.13	69.98
7	10	79.25	73.18	69.03
8	10	79.25	70.97	65.84
9	8	78.28	69.94	64.86
10	8	78.28	72.71	66.39
11	8	78.28	71.87	64.32
12	6	77.03	74.36	66.36
13	6	77.03	71.40	61.36
14	6	77.03	65.03	52.85
15	4	75.27	75.73	62.60
16	4	75.27	76.20	59.40
17	4	75.27	76.77	55.22
18	4	75.27	62.77	36.33
19	2	72.26	62.87	31.55
20	2	72.26	61.71	23.19
21	2	72.26	60.68	13.46
22	2	72.26	59.77	0.77
23	2	72.26	60.04	−16.13
Overall	178	91.75	91.62	88.51

SWL In: source Sound Power Level; SWL Out: transmitted Sound Power Level.

This prediction of attenuation using only the axially symmetric mode is similar to Rice’s modeling of broadband noise with the radial modes required to synthesize the plane axisymmetric mode¹¹ at the source plane of the semi-infinite lined duct. However, there are significant differences in the models. Rice’s source is deterministic, based on representing the plane wave at the entrance to the lining, while the source model here is statistical. Additionally, the current model has scattering associated with a finite length lining and hard wall source and termination sections.

Results for all 6 propagating circumferential modes with 16 propagating radial modes are summarized in Table 5. The table organization was discussed in connection with Table 4, but now shows the six propagating circumferential modes, each with their propagating radial modes (including $\pm m$ for $m > 0$ ). The reference $SIL$ for each circumferential mode, as distributed by the weighting factors of equation (33), is seen to reflect the number of propagating radial modes in the circumferential mode. Input SWL and transmitted SWL, as reconstructed from the statistical propagation solutions, are tabulated. Also shown are the overall results for all propagating modes. Overall, input SWL now represents the contributions from six circumferential modes that are individually driven by reference Sound Intensity Levels distributed according to the third column. Overall attenuation is almost 8 dB, 2.4 dB higher than predicted using only the axially symmetric mode (about 43% higher than the axially symmetric approximation). This result at $450 Hz$ suggests that at this frequency the axially symmetric circumferential mode alone as a model for broadband noise is not a close predictor of attenuation. The distribution of input SWL here is relatively uniform and not necessarily always the case. In fact, it is this relatively uniform distribution, with well attenuated higher order circumferential modes, that results in the substantially higher overall attenuation in the case modeled with all six propagating circumferential modes as compared to the axially symmetric model.

In the higher frequency case, $f = 1650 Hz$ , the summary of Table 6 is given for the axially symmetric model. This shows the single incident circumferential mode $m = 0$ , with the count of the propagating radial modes at 8. This circumferential mode accounts for the entire overall input and transmitted power, and therefore the entire attenuation in acoustic power. Attenuation of acoustic power is approximately 2 dB, considerably lower than achieved in the $450 Hz$ case. As discussed in the lower frequency example shown in Table 4, this case can be considered an analog to Rice’s axially symmetric model for broadband noise.¹¹

When all propagating circumferential modes are included in the $1650 Hz$ case, Table 7 summarizes the results for the 24 propagating circumferential modes and the corresponding 178 radial modes (including $\pm m$ for $m > 0$ ). Overall attenuation is about 3 dB, about 1 dB higher than the axially symmetric prediction (about 50% greater than the axially symmetric approximation). The axially symmetric mode alone is still not a good predictor of attenuation of broadband noise.

As in the lower frequency case, reference $SIL$ in the circumferential modes is distributed according to equation (33), and the influence of the weighting factors is apparent. Contrary to the observation in the $450 Hz$ case, the distribution of input SWL in the 24 circumferential modes is not nearly uniform and is dominated by the $m = \pm 1$ and $m = \pm 2$ spinning modes. This is a result of the statistically based propagation solutions and the resulting acoustic pressure fields and is not a general result. Even though acoustic power attenuation increases rapidly with increasing circumferential mode order, the decreasing contribution of the distributed higher order modes to overall SWL makes the lower order, less heavily attenuated circumferential modes dominant in determining overall attenuation.

Acoustic power attenuation is much lower in the $1650 Hz$ case, than in the $450 Hz$ case. This is predicted using either the axially symmetric model or the all propagating circumferential mode model. It is not possible to draw a general conclusion from this result because lining impedance is a determining factor in lining performance. The two degrees of freedom lining in the present example was optimized for tones at $m = 11$ , $1100 Hz$ ( $η = 14.3$ ) and $m = 22$ , $2200 Hz$ ( $η = 28.6$ ); $450 Hz$ and $1650 Hz$ played no role in lining optimization. At $450 Hz$ , $α_{0} = 6.99$ , and the range of the parameter $\frac{m}{α_{0}}$ is as high as 0.86 at $m = 5$ and at $1650 Hz$ , with $α_{0} = 25.65$ , $\frac{m}{α_{0}}$ is as high as 0.90 at $m = 23$ . Figure 1, from Rice’s estimation formula,¹⁰ shows that in either case the high-order circumferential modes are very nearly cut-off. Note that the validity of Rice’s approximation is being pushed in the $450 Hz$ case with $α_{0} = 6.99$ . These high-order circumferential modes can be expected to be scattered into highly attenuated circumferential modes in the lined section of the duct. This is verified in Tables 5 and 6. The highly attenuated high-order circumferential modes may or may not contribute strongly to overall attenuation, depending on the distribution of power among the circumferential modes. In the $450 Hz$ case, with relatively few propagating radial modes in all of the circumferential modes, the distribution is relatively uniform. Attenuation of the higher order circumferential modes contributes significantly to overall attenuation. At $1650 Hz$ , with many propagating radial modes in many of the circumferential modes, the distribution is not uniform, and favors the lightly attenuated lower circumferential order modes. This leads to little influence of the highly attenuated high-order circumferential modes on overall attenuation, leaving the lightly attenuated low-order circumferential modes to dominate.

These observations are relevant only for the statistically based approach to distributing acoustic power among the circumferential modes from a presumed source definition with an $SIL$ spectrum. A second approach, outlined in equations (26) to (29), based on source definition in terms of a Sound Power spectrum, removes some of the statistical nature of the distribution of power among circumferential modes by relating the power in each mode directly to the number of propagating radial modes in the mode, and bypassing the scaling of statistical propagation solutions to obtain the distribution of power among the circumferential modes. In this investigation, the $SIL$ spectrum approach has been used because it interfaces better with data likely to be available.

Discussion

Frequency domain methods provide an effective means to model propagation of broadband noise in turbofan inlet ducts. Perhaps, most importantly, frequency domain techniques preserve the usefulness of the work of Rice^10–17 that has (1) provided for the estimation of the number and distribution of propagating modes, enabling the formalization of the suggestion of Dyer⁵ that broadband noise consists of all propagating modes with equal power and random phase and (2) connected the concept of cut-off ratio to propagating modes in unlined ducts and to attenuation in lined ducts. In the present investigation, numerical schemes have been advanced to the extent that what might have been achieved laboriously by serial implementation of Rice’s computational approach, and graphical presentation is now efficiently done by FEM frequency domain modeling with codes written in parallel architecture. The approach employed here has been to extend an FEM code written in serial architecture, capable of modeling propagation at a single frequency and circumferential mode, to a parallel architecture that in the present application models all propagating circumferential modes for a specified frequency. This enables the implementation of the representation of broadband noise proposed by Dyer.⁵

Broadband noise has been represented in the frequency domain by a statistical distribution of propagating duct modes. The required number of modes has been estimated by Rice’s approximation,¹⁰ backed up by exact calculations in the FEM propagation code. Two models for estimating achievable attenuation for broadband noise attenuation have been examined. One uses only the axially symmetric circumferential mode, suggested by Rice’s study of broadband noise.¹¹ A second approach uses all propagating circumferential modes and their associated propagating radial modes. The parallel architecture code is designed to include all propagating modes on a statistical basis. Examples for prediction of acoustic power attenuation of broadband noise at low frequency and at moderate frequency have been used to compare the acoustic power attenuation prediction performance of the axially symmetric circumferential mode and the all propagating circumferential mode models. Computational demands increase with the number of circumferential modes required to include all propagating radial modes, but only to the extent of the number of processors used.

There are differences in the propagation model used by Rice,¹¹ and the one used here. In Rice’s axially symmetric model, radial mode content is deterministic, with mode amplitudes chosen to represent a plane wave at the entrance to the lining. In the present model, the lining is imbedded in a duct with hard wall source and termination sections that produce scattering of statistically distributed hard wall duct modes into the lined section, and scattering of lined duct modes into hard wall duct modes in the termination section. The model used here with the source represented by a statistical distribution of hard wall duct modes makes direct use of Rice’s modal density work.¹⁰

Broadband propagation at two frequencies, one low and one moderate, has been investigated to assess how the axially symmetric model and the multiple circumferential mode model compare for prediction of acoustic power attenuation. It has been found that the axially symmetric approximation predicts lower attenuation, as compared to the all circumferential mode model in the both cases, and it is not a good predictor of attenuation when compared to the all propagating circumferential mode model. Both models capture the fact that attenuation is higher at the lower frequency. Rice’s correlation of mode cut-off ratio with propagation and radiation^15–17 can be used to rationalize this observation.

All conclusions reached in this investigation have been obtained with distribution of circumferential mode power based on Sound Intensity Level measurements defining the source. A second approach with source data defined by SWL measurements leads to a more predictable distribution of acoustic power among the circumferential modes, but may not be as useful due to the unavailability of this type of data.

The axially symmetric model, originally suggested by Rice,¹¹ remains relevant. It accurately predicts the trend of achievable attenuation with frequency, though with lower attenuation than predicted by multiple circumferential modes.

An appreciation

It is a privilege to participate in this issue honoring the contributions of Ed Rice to the field of aeroacoustics, and a particular privilege to recognize the huge influence he still has on the way we approach the design of acoustic linings for the control of turbofan noise. His work identifying the importance of cut-off ratio of duct modal propagation in the prediction of acoustic lining performance, and his modeling of the impedance of perforates, were seminal. Ed’s remarkable physical insight allowed him to analyze complex phenomena in duct acoustics when computational power was far less than today. It is fitting that this volume has been compiled from authors whom he has influenced.

Footnotes

Declaration of Conflicting Interests

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

Funding

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

Appendix

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