Time-Domain Simulation of Nonlinear Acoustic Propagation in a Lined Duct

(1)

American Institute of Aeronautics and Astronautics 1

Time-Domain Simulation of Nonlinear Acoustic Propagation in a Lined Duct

Chi-Hoon Cho^* and Duck-Joo Lee.^† KAIST, Daejeon, Korea

and

Cyrille Breard^‡, John Premo^§, Justin H. Lan^**

The Boeing Company, Seattle, WA 98124

The nonlinear behavior of finite amplitude acoustic waves, such as attenuation through shocks and energy transfer between frequencies, are important considerations in the prediction of the inlet fan noise. In this study, a time-domain nonlinear acoustic propagation code with an impedance boundary condition representing the acoustic liners is presented.

The full Euler equations are utilized as a model equation for nonlinear wave propagation.

Spatial discretization of the equations is accomplished using a high-order conservative scheme on a cell-centered finite volume grid. A time-domain impedance boundary condition is implemented based on a characteristic boundary approach, which is equivalent to the use of a reflection relation in the linear case. The order of accuracy of the broadband impedance model using the z-transform is improved over previous implementations. The NASA Langley grazing impedance tube configuration with a ceramic tubular (CT) liner is adopted for the nonlinear simulation of a lined duct. For the nonlinear simulations, impedance non- linearity is ignored and only propagation nonlinearity is considered. Simulations of nonlinear propagation for two types of source waveforms are conducted. For saw-tooth wave sources, the energy transfer between the harmonic components is clearly observed during the propagation in the lined duct. For sinusoidal sources at 1 kHz, the attenuation rate by the acoustic liner at the fundamental frequency is the same regardless of source amplitude.

Nonlinearity of the liner is discussed by comparison with experimental data.

Nomenclature

et = specific total energy

i = imaginary number

p = pressure

p' = pressure perturbation (= p − p¶)

ρ = density

SPL = sound pressure level

t = time

(u ,v) = velocity

vn = flow velocity component normal to the surface x, y = Cartesian coordinates

ω = angular frequency

z = z-transform variable

Z = impedance

* Graduate Student, Division of Aerospace Engineering

† Professor, Division of Aerospace Engineering, Senior Member AIAA

‡ Scientist/Engineer, Acoustics & Fluid Mechanics Technology, MS 67-ML, Senior Member AIAA

§ Now in GE Aviation, Subsection Manager Acoustics and Installation Aerodynamics, Cincinnati. Senior Member AIAA

** Engineer, Acoustics & Fluid Mechanics Technology, MS 67-ML, Senior Member AIAA 14th AIAA/CEAS Aeroacoustics Conference (29th AIAA Aeroacoustics Conference)

5 - 7 May 2008, Vancouver, British Columbia Canada

AIAA 2008-2830

(2)

I. Introduction

The sound pressure level (SPL) of inlet fan noise generated from aero-engines can exceed 160 dB near the fan face.

At this high SPL, nonlinear phenomena such as energy transfer between frequency components during the propagation are important in the prediction of noise propagation. For example, the propagation characteristics of multiple pure tone (MPT) or ‘buzzsaw’ noise is highly nonlinear¹. In this paper, the authors consider nonlinear acoustic propagation in a duct with and without acoustic lining.

When an acoustic wave with high amplitude propagates through air, its waveform becomes distorted. This phenomenon is called the ”wave steepening” effect². In the frequency domain, the effect leads to the generation of harmonic frequency components. Wave steepening may also lead to weak shock formation. Therefore, the requirements for the numerical simulation of the nonlinear wave are i) shock-capturing and ii) simultaneous multi- frequency simulation.

Most CAA schemes are based on central differencing, which requires some method such as artificial dissipation to prevent numerical instability. However, for non-linear propagation it can be difficult to control the spurious instability near shocks by artificial dissipation in a high-order central differencing scheme. In this study, a high- order upwind monotonicity-preserving (MP) scheme is used because of its stability and robust shock-capturing ability without introducing artificial dissipation.

The effect of the acoustic liner is modeled through a broadband impedance boundary condition. Impedance is usually modeled and measured in the frequency domain and is not directly used in the current time-domain simulation. A characteristic-based impedance boundary treatment applicable for broadband frequency is presented.

Based on the work of Özyörük et al.³, the time-domain impedance boundary condition is implemented within the framework of characteristic variables.

The NASA Langley grazing impedance tube configuration⁴ shown in Fig 1 is adopted for the lined duct simulation. Linear simulation of a lined duct without flow is conducted for validation of the present impedance boundary condition. The simulated data are compared with the experimental data.

Nonlinear cases are simulated by using the Euler equations. Two kinds of source waveforms, sinusoidal and saw-tooth, are considered to investigate the various features of the nonlinear propagation in the duct. Numerical solutions are compared with analytical solutions obtained by using weak shock theory, for a hard wall straight duct.

For a lined duct, the nonlinear results are compared by varying the source amplitude from 120 dB to 160 dB. For high SPL, the results are also compared with a nonlinear propagation code from Boeing. Nonlinear propagation in a lined duct is discussed by comparing simulated results with a sinusoidal source with experimental data.

II. Numerical Methods A. Governing Equations and Discretization Methods

For linear propagation the isentropic linearized Euler equations (LEE) for zero mean flow are used, and for nonlinear propagation the full Euler equations are used. The conservative form of the linear (Eqn 1b) and non-linear (Eqn 1c) equations are;

= 0

∂ + ∂

∂

y x t

F E

Q

(1a)

⎥ ⎥

⎥

⎦

⎤

⎢ ⎢

⎢

⎣

⎡ ′

= v u p Q

,

⎥ ⎥

⎥

⎦

⎤

⎢ ⎢

⎢

⎣

⎡

= ′ 0 p u E

,

⎥ ⎥

⎥

⎦

⎤

⎢ ⎢

⎢

⎣

⎡

′

= p v

F 0

^{, (1b)}

⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

⎡

= e

t

v u ρ

ρ ρ ρ Q

,

⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

⎡

+

= +

pu u e

vu p uu

u

ρ

t

ρ ρ

ρ

E

,

⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

⎡

+

= +

pv v e

p vv

uv v

ρ

t

ρ ρ

ρ

F

. (1c)

(3)

Since the 1990’s, numerical solutions of the linear equations for aeroacoustic applications have favored dispersion-relation-preserving (DRP) and compact finite difference schemes^5-8 because of their accuracy for linear acoustic wave propagation. However, the schemes have difficulties with simulating problems containing shock phenomena, because their artificial-dissipation schemes are non-conservative central schemes. For the problems with shock phenomena, in principle, conservative or shock-capturing schemes should be utilized to suppress the numerical oscillation near the shock and to capture the shock position correctly.

In this study, a high-order conservative scheme is used. To make the numerical solver conservative, a cell- centered finite volume grid is employed. Conservative variables such as mass and momentum in a grid cell volume are calculated for the representative values of each cell center, whereas the flux variables are allocated at the cell faces.

The monotonicity-preserving (MP) spatial discretization scheme developed by Suresh et al.⁹ is used for calculation of the cell-face fluxes with high-order accuracy and with limitation in MP criteria. In general, MP schemes show less numerical dissipation than the weighted essentially non-oscillatory (WENO) fifth-order scheme by Shu et al.¹⁰ which uses total variation bounded (TVB) criteria. To apply this spatial scheme to a conservative system, the following upwind process is employed. Flux variables located at cell-centers are split to local Lax- Friedrich (LLF) fluxes and then the split fluxes are projected to characteristic variables, which are interpolated at cell faces by the MP scheme. Interpolated characteristic variables are then projected to LLF fluxes and are merged to produce the resultant numerical fluxes at the cell-face.

An explicit strong stability-preserving (SSP) Runge-Kutta third order method¹¹ is used for the time-marching scheme.

A hard wall boundary condition is imposed by using an image ghost cell, and the acoustic source and nonreflecting boundary conditions are implemented by characteristic boundary conditions.

B. Time-domain Impedance Boundary Condition (TIBC)

The impedance for locally reacting liners is usually defined in the frequency domain as the following;

) ( / ) ( )

(ω p ω v_n ω

Z = ′ (2)

Several techniques^3,12-14 to impose this condition on the acoustic liner boundary in a time-domain numerical solver have been published. Some important considerations are how to embed the boundary condition with numerical stability and accuracy and how to transform the impedance values defined in the frequency domain into the time- domain. In this study, a characteristic boundary condition is used which is similar to the one from Fung et al.¹³ and also the z-transform technique for a time-domain transformation of a broadband fitting function proposed by Özyörük et al.³

1. Characteristic TIBC

The characteristic time domain impedance boundary condition for an acoustic liner is presented. Consider the following one-dimensional characteristic y-directional equation;

0 0

0

0 0

=

⎥⎥

⎥

⎦

⎤

⎢⎢

⎢

⎣

⎡

∂

⎥⎥

⎥

⎦

⎤

⎢⎢

⎢

⎣

⎡

− + +

⎥⎥

⎥

⎦

⎤

⎢⎢

⎢

⎣

⎡

∂

− +

s

s L

L L v y c v c v

L L L

t ⁽³⁾

ρ ρ

ρ ∂ ∂ =∂ − ∂ ∂ =∂ − ∂ +

∂

=

∂ ₊ , ₋ , and ²

where L p c v L p c v L_s p c ^.

With the isentropic assumption, the entropy wave Ls is zero. Since the acoustic wave variables L+ and L− are defined as a differential form such that it cannot be found explicitly, the local linearization with the appropriate frozen state (index f) is applied. After integration, the variables can be expressed as

v c p L

f f

) (

ρ ρ

′−

=

′+

=

−

+ (4)

(4)

Considering the impedance boundary located at the +y direction, L+ and L− are the outgoing acoustic wave and incoming acoustic wave through the boundary, respectively. The outgoing wave value at the boundary can be calculated by a one-sided Lagrange interpolation from the nearest boundary stencil as shown Fig. 2. With the impedance relation Z = p' / v, we can derive the following relation between the wave variables;

+

− = −

+ c L Z c L

Z( ) ( )_f) ( ( ) ( )_f)

( ω ρ ω ρ (5)

This relation is used to obtain the unknown L− from the known L+ . The frozen state ρf and cf is set to the boundary cell value. If the states are constant, for example, in linear simulation, Eq. (5) becomes the same to the wave reflection relation which is utilized in the work of Fung et al. Selection of the Z(ω) function and the technique for solving Eq. (5) are described in the next section.

2. Impedance Fitting Function and z-Transformation

To transform the frequency domain equation to the time-domain, Z(ω) needs to be a continuous function of ω.

Since the impedance data are measured in the discrete frequency domain in experiments, a continuous function which fits the discrete data within a certain bandwidth should be introduced. Selection of the fitting function with the optimal number of coefficients depends on the shape of the impedance data. One of the simplest fitting functions is a three-parameter spring-mass-damper type function:

)

( ₁ ₂ ³

ω ω

ω i

r r i r

Z = + + ⁽⁶⁾

which generally fits narrow band or off-resonance bands. Using this function, the impedance relation can be transformed to the time-domain with temporal derivative and integral terms. To handle more general or broader bandwidth, the function needs to have more parameters with a well-chosen shape function. Özyörük et al.³ suggested a fitting function for the ceramic tubular liner tested by NASA, which has an additional band-pass filter type term.

2 6 5

4 7

3 2

1 1 1 ( )

)

( i r i r

r r i

r i i r r

Z ω ω

ω ω ω ω

+ + +

= ⁽⁷⁾

The coefficients ri should be positive values, and the poles of iω should have non-positive real-values. In this paper, CT57 and CT73 liners tested by NASA Langley are selected for the liner because their experimental data are available and it has been known that the impedance nonlinearity of the liners can be ignored⁴. Performance of the liners can be roughly estimated by the schematic figures shown in Fig. 3(a). The liner consists of an array of tubes with a 85.6 mm depth. In a tube, longitudinal standing waves are generated at several frequencies. Experimental impedance data and their fitting curves for CT57 and CT73 liners are plotted in Fig. 3(b). The coefficients are calculated to fit the impedance data of 130 dB in Table 1 by using a nonlinear optimization routine in a MATLAB program. The resulting coefficients are shown in Table 2.

The impedance fitting function is substituted into Eq. (5). Then, the equations can be solved in the discrete time- domain using a z-transform. The z-transform gives an approximate transformation between the z-domain (frequency) and the discrete time domain. A discrete Fourier transform is related to the z-transform as follows:

)

1 exp( i t

z⁻ = −ωΔ (8)

To utilize the time-shifting property of the z-transform, Eq. (8) can be approximated by the Taylor or Padé series expansion at z⁻¹=1. In Ref. 3, Özyörük et al. used the first-order backward approximation,

t i z

Δ

=1− ⁻¹

ω ⁽⁹⁾

however, there were discrepancies between the broadband and single frequency simulations. In the present study, a second-order backward difference (BDF2) approximation is used.

(5)

t z i z

Δ +

= − ⁻ ⁻

2 4

3 ¹ ²

ω ⁽¹⁰⁾

After substituting Eq. (10) into Eq. (7), the z-domain representation of impedance becomes the ratio of :

∑ ∑

−

= _m

m m m

z b

z z a

Z( ) ⁽¹¹⁾

With Eq. (11), the wave variable relation (Eq. (5)) becomes

) ( ) )

( (

) ( ) )

(

∑

a_mz⁻^m⁺ ^ρc _f

∑

b_mz⁻^m L₋ z ⁼

∑

a_mz⁻^m⁺ ^ρc _f

∑

b_mz⁻^m L₊ z ⁽¹²⁾ After applying the time-shifting property and arranging, the final equation for unknown L− is obtained as;

( )

∑

^∞

=

−

− +

−

− + +

− − − +

+ + +

= −

0 1 0

0 0

0

0 ( ) ( ) ( )

) (

1 )

( ) (

m

m n m n m f m

n m n m f

n f n f

L L

b c L

L b a

c L a

b c a

b c

L a ρ

ρ ρ

ρ . (13)

III. Validations

NASA Langley GIT data are used as validation. Before discussion of the simulation results, the grid and computational cost for the simulations are briefly noted. The grid spacing in both x- and y- directions is 1/30 of the 2-in duct height, which corresponds to 405 number of points per wavelength (NPPW) at 0.5 kHz. The total number of grid points is approximately four thousand. Computational wall clock time per 100 time steps is approximately 10 seconds with a code using OpenMP on two 1.2GHz Intel Core2 machines. After about 10000 steps, the numerical solution becomes temporally periodic.

The numerical simulations for the CT liner with no flow are performed. LEE equations are used as the governing equations. A nonreflecting boundary condition is used at the exit and a transparent source boundary condition at the inlet. The speed of sound is set as 343 m/s for all present simulations.

A. Multi-frequency Simulation

Results from simulations with several simultaneous frequencies are compared with those from the corresponding single-frequency simulations. For the impedance, the broadband fitting function (Eq. (7)) is used. Here, the single- frequency simulation means that the acoustic source has a single frequency and the impedance boundary condition with a second-order approximation z-technique is used. For multi-frequency simulation, the acoustic source has many frequency components, and the broadband fitting function (Eq. (7)) with z-transform technique is used. For both cases, the time-domain pressure data at the wall surface are Fourier-transformed, and the SPL of each frequency component is plotted. As shown in Fig. 4(a), if the first-order approximation is used in the z-transform, the multi-frequency simulation shows some discrepancies relative to the single frequency simulation, which can also be found in Ref. 3. However, by using the second-order approximation, the broadband simulation is more accurate as seen Fig. 4(b).

B. Comparison with GIT Experimental Data

Linear simulation results for the CT57 liner are validated by comparing them with the NASA Langley GIT experimental data for a zero-mean flow case. SPL data along the hard wall are shown in Fig. 5 from 500Hz to 3 kHz.

Although the experimental work suggested using the measured impedance condition for the exit plane, a nonreflecting condition is assumed in the simulation because the measured impedance is almost 1+i0 which is equivalent to the non-reflecting condition. Good agreement between them is seen except in the 1 kHz case. A notable feature of the 1 kHz case is that a remarkable amount of noise attenuation occurs. There are some discrepancies in attenuation slope and noise level after the liner region, which are also found in the result of Ref. 16.

(6)

IV. Nonlinear Simulation Results

For a linear acoustic model such as a linear scalar wave equation or the LEE, the principle of superposition is valid, which means that the frequency components are independent of each other. But air has inherent nonlinear characteristics, which become significant when the amplitude of the wave is sufficiently high. Mathematically, the nonlinearity originates from the nonlinear convection term in the momentum equation and the nonlinear relation between density and pressure in the state equation. With a nonlinear model such as the full Euler equations, the principle of superposition is not valid. The resulting change due to the nonlinearity will depend on amplitude, frequency, propagation mode and other factors. In this section, nonlinear propagation of a high amplitude wave through the lined duct is investigated by numerical simulation of the full Euler equations. Two kinds of source waveforms, sinusoidal and saw-tooth, are considered in investigating the various features of the nonlinear propagation in the duct.

A. Comparison with Analytic Nonlinear Solution for Hard-Wall Duct

We consider two examples of nonlinear plane-wave propagation, whose analytic solutions exist, for the purposes of code validation and brief review of nonlinear propagation. The Fubini solution is the analytic solution for the sinusoidal waveform source². It is valid over the region from the source position to the shock-formation position. If the acoustic wave contains a weak shock, the weak shock theory can be applied to specify its location¹⁷.

The nonlinear propagation of an initially sinusoidal wave is simulated numerically. Results are presented by the graph of SPL vs. distance for each frequency component as shown in Fig. 6(a). Amplitude levels at 140 dB and 160 dB are used with a fundamental frequency of 1 kHz. In the figures, the higher harmonic components beyond the sixth are omitted or out of the plotting range. For both amplitude cases, numerical solution with NPPW=202 is in excellent agreement with the Fubini solution. For 140 dB, the generation of the second harmonic component is 107 dB which is 3 % of total amplitude 140 dB at x=40 inch while, for 160 dB case, it is 146 dB, which corresponds to 20 % of total amplitude. The amplitudes of higher harmonics also have comparable magnitudes.

For the initially saw-tooth waveform type, the waveform shape does not change but the amplitude decays slowly by the physical non-linear decay at the shock. The numerical solutions with NPPW=202 and the analytic saw-tooth solutions are compared in Fig 6(b). The fundamental frequency is also 1 kHz. This figure shows good agreement for the fundamental and second harmonic components. The numerical results show slightly smaller amplitudes compared to analytical saw-tooth solutions due to numerical dissipation near the source position. The maximum error is less than 0.5 dB at x=70 inch.

B. Saw-tooth Wave Source for Lined Duct

Due to the wave steepening effect in nonlinear propagation, a high amplitude acoustic wave, radiated from a temporally periodic source, usually evolves to have a saw-tooth-like waveform.. In this section, the nonlinear propagation of saw-tooth waves through the lined duct is analyzed. For the liner, the fitted impedance data of CT liners are used. The frequencies of high harmonic components in the saw-tooth waveform can be out of the measurement frequency range. At these frequencies, the present broadband impedance model imposes the extrapolated impedance values from the fitting function. The fundamental frequency is 1 kHz. The nonlinearity in propagation depends on the amplitude level of the wave. The amplitudes of 120 dB, 140 dB and 160 dB represent linear, moderately nonlinear and strongly nonlinear propagation, respectively.

Figure 7 shows the simulation results for CT57 liner. The case of 120 dB is considered to be linear and is used as the baseline for comparison. The SPL attenuation slopes for the nonlinear cases (140 dB, 160 dB) change slightly from the linear case over the liner region. However, it is observed that for 140 dB the amplitude of the fundamental component decreases after the liner, but for 160 dB, the amplitude increases rapidly. From Fig. 7 (d), the difference between the linear case and the strongly nonlinear case are clearly observed. The difference might be explained by the relatively high amplitudes of all harmonics for the 160dB case after the liner section (140-145dB). For the 140dB case, amplitudes of all harmonics are below 130 dB and are nearly constant in the hard wall section after the liner, thus wave propagation is considered to be in the linear regime. The waviness of the 4^th, 5^th and 6^th harmonics are due to an interference pattern with transverse modes that are cut-on at that frequency.

(7)

C. Sinusoidal Wave Source for Lined Duct

Sinusoidal waves are specified at the source plane of the lined duct in this section. The broadband fitting function (Eq. (7)) with z-transform technique is used to provide the impedances values at the CT73 liner surface.

Like the saw-tooth simulations, the source frequency 1 kHz is used. The results are shown in Fig. 8.

The attenuation slope for the source frequency over the liner is not changed for the 140 dB and 160 dB cases.

Whether the case is linear or nonlinear, attenuation remains the same with the same impedance values. This finding is from artificial conditions because we use the same impedance for different source levels, but conversely it can be stated that if the slope of the experimental data is the same, the impedances are the same.

For the nonlinear cases harmonic components are generated during propagation. For 160 dB, the third harmonic component is also shown in the figure. The generation of the harmonic components is purely due to the nonlinearity in the propagation. When the liner is absent, the nonlinear propagation of the harmonics is shown as the dotted curves in the figure. The generation of the harmonic components occurs in the region from acoustic source to the leading part of liner, where the levels are highest. Accordingly, if the distance between the acoustic source and the liner increases, the amplitude of each harmonic will increase, proportionally. After the liner region, the overall SPL decreases to linear levels and the energy exchange between the harmonics does not occur, so that, the amplitudes of these frequency components are constant.

Due to the generation of the harmonics by nonlinear propagation, the overall performance of the liner can be controlled by the harmonics when the liner has effective performance for the fundamental component. For the 140 dB case, the amplitude of the second harmonic behind the liner region is comparable to that of the fundamental frequency component. For the 160 dB source, the second harmonic amplitude exceeds that of the fundamental by about 20 dB. In Fig. 8(d), the time signals at x=0, 16 and 40 inch are plotted. At the source position of x=0, the signal is 1 kHz, but the second harmonic frequency 2 kHz dominates the signal at x=40 in. after the liner.

From the previous simulation results for the sinusoidal sources, we conclude that if the impedance values for the acoustic liner are retained regardless of the source levels, the attenuation slope for the source frequency is barely changed. However, for source amplitudes greater than 140 dB, the propagation nonlinearity appears as generation of the second harmonic frequency component.

In the GIT experiment conducted by NASA, the source levels are 130 dB or 140 dB. From the viewpoint of the propagation nonlinearity, 130 dB corresponds to a marginally nonlinear case and 140 dB corresponds to a moderately nonlinear case. If there is no difference between the fundamental frequency data for 130 dB and 140 dB in the experiment, the nonlinearity in the acoustic liner is not important whether the nonlinearity in propagation exists or not. Figure 9 shows the comparisons between simulation and experimental data for 1 kHz for 130 dB and 140 dB. In the experimental data, the attenuation slope for 130 dB and 140 dB sources are almost the same. It means that for CT57 liner the impedance nonlinearity can be ignored and the impedance value is not changed at least until the source level reaches 140 dB. In the simulations, the educed impedance of CT57 liner for the 130 dB source (Z=0.46+i0.00) is used for both amplitudes although the educed impedance for 140 dB (Z=0.49+i0.12) is different from the one used for 130 dB. There is good agreement for the fundamental frequency between the simulations and experimental data for the 130 dB and 140 dB sources. In Fig. 9, the second harmonic components from the simulation are also provided. The nonlinearity is clearly shown with the appearance of harmonic components in the simulation, whereas the nonlinearity in the experimental data cannot be shown if only source-frequency data are measured by FFT from the time signal. Therefore, experimental data for higher harmonics is definitely necessary for even a source level as low as 140dB. If experimental data from higher harmonic components was available, it would be possible to analyze the propagation nonlinearity occurring in the experiment and also the data could be used for the validation of the nonlinear code, as reported in the next section.

D. Validation with nonlinear frequency domain code

Predictions for a saw-tooth type wave with the Boeing non-linear code LEE2D¹⁶ has been performed for the case corresponding to Figure 7 (b) but with acoustic liner CT57. Only 6 harmonics were included in the computation.

The fitted impedance data of the CT57 liner was used since no data are available for frequencies higher than 3000Hz which corresponds to the 3^rd harmonic. The results are compared with those obtained with the present KAIST code in Figure 10. Agreement between the hybrid-frequency domain (Boeing) and the time domain (KAIST) approaches are excellent. In addition, both codes captured the scattering of higher transverse modes, as shown by the interference pattern for the 4^th, 5^th and 6^th harmonics. Good agreement between the two codes reinforces the indication of correct implementation and interpretation of the results.

(8)

V. Concluding Remarks

A time-domain nonlinear propagation code with acoustic liner was developed and applied to the simulation of propagation in lined ducts. An upwind finite volume method using an MP scheme is used to capture nonlinear waves with high accuracy. A time-domain impedance boundary condition is implemented by using a characteristic approach. To apply the code for broadband frequencies, a second order accurate z-transform is used. Validations of the impedance boundary conditions were performed using experimental data and showed good agreement.

For the nonlinear cases, sine and saw-tooth source waveforms are simulated. The nonlinear propagation in the duct without the liner is simulated to validate the code with analytical solutions. For nonlinear propagation in a lined duct with the saw-tooth wave source, the energy exchange between the frequency components is observed. From the simulation results with a single frequency source, two interesting features are found: 1) The attenuation for the source frequency is barely changed with source amplitude level when the liner impedance is assumed linear. This implies that when the measured attenuation does not change even at sufficiently high source levels, the nonlinearity in impedance is not important below a source level of 140dB. 2) Nonlinear propagation appears as generation of harmonic components whose amplitudes may exceed the amplitude in the source frequency.

Due to the lack of experimental data with very high sound pressure level, results from the time-domain (KAIST) code have been compared with those from the frequency-domain (Boeing) code. Good agreement between the two codes has been shown.

In the future, both codes could be used for the prediction of nonlinear propagation in engine inlet duct geometries at realistic flow conditions accounting for spinning mode sources.

Acknowledgments

The first two authors would like to acknowledge Dave H. Reed and Michael J. Czech for arranging the financial support from the Boeing Company for this work.

References

[1] A. McAlpine and M.J. Fisher, "On the Prediction of Buzz-Saw Noise in Aero-Engine Inlet Ducts", Journal of Sound and Vibration, Vol. 248, No. 1 (2001)

[2] D.T. Blackstock, M.F. Hamilton, A.D. Pierce, “Ch4. Progressive Waves in Lossless and Lossy Fluids” in Nonlinear Acoustics, edited by MF Hamilton and DT Blacstock, Academic Press (1998)

[3] Y. Özyörük, L. N. Long, M.G. Jones, Time-Domain Numerical Simulation of a Flow-Impedance Tube, J. Comput. Phys. Vol.

146, 29 (1998)

[4] M. G. Jones, W. R. Watson, and T. L. Parrott, “Benchmark Data for Evaluation of Aeroacoustic Propagation Codes with Grazing Flow”, AIAA Paper No. 2005-2853 (2005)

[5] J.C. Hardin, J. R. Ristorcelli and C. K. W Tam (editors), “ICASE/LaRC Workshop on Benchmark Problems in Computational Aeroacoustics (CAA)”, NASA CP-3300 (1994)

[6] Tam, C.K.W. and Hardin, J.C. (editors), “Second Computational Aeroacoustics (CAA) Workshop on Benchmark Problems”, NASA CP-3352 (1997)

[7] M. D. Dahl (editor), “Third Computational Aeroacoustics (CAA) Workshop on Benchmark Problems”, NASA CP-2000- 209790 (1999)

[8] M. D. Dahl (editor), “Forth Computational Aeroacoustics (CAA) Workshop on Benchmark Problems”, NASA CP-2004- 212954 (2004)

[9] A. Suresh, H.T. Huynh, Accurate Monotonicity-Preserving Schemes with Runge–Kutta Time Stepping, Journal of Computational Physics, Vol. 136, No. 83 (1997)

[10] G.S. Jiang and C.W. Shu, "Efficient implementation of weighted ENO schemes", Journal of Computational Physics, Vol.126 (1996)

[11] C.W. Shu and S. Osher, "Efficient implementation of essentially non-oscillatory shock-capturing schemes", Journal of Computational Physics, Vol.77 (1988)

[12] C. K. W. Tam and L. Auriault, Time-domain impedance conditions for computational aeroacoustics, AIAA J. 34(5), 917 (1996).

[13] Fung, K.-Y., and Ju, H., “Impedance and Its Time-Domain Extensions,” AIAA Journal, Vol. 38, No. 1 (2000) [14] Fung, K.-Y., and Ju, H., “Broadband Time-Domain Impedance Models,” AIAA Journal, Vol. 39, No. 8 (2001)

[15] J.H. Lan, C. Breard, “Development and Validation of a 3D Linearized Euler Solver”, AIAA Paper No. 2006-2585 (2006) [16] C. Breard, “A frequency-domain solver for the nonlinear propagation and radiation of fan noise” , Unsteady aerodynamics, aeroacoustics and aeroelasticiy of turbomachines (2003) pp.275-289

[17] D.T. Blackstock, “Connection between the Fay and Fubini Solutions for Plane Sound Waves of Finite Amplitude,” Journal of the Acoustical Society of America, Vol. 39, No. 6, (1966)

(9)

130 dB source, M=0.0 140 dB source, M=0.0 f [Hz]

R X R X

500 0.51 -1.68 0.58 -1.74

1000 0.46 0.00 0.49 0.12

1500 1.02 1.30 1.16 1.27

2000 4.05 0.62 5.01 0.96

2500 1.54 -1.60 1.49 -1.59

3000 0.70 -0.29 0.74 -0.19

Table 1. Educed Impedance of CT57 Liner

CT73 liner ¹⁹ CT57

r1 = 3.468881408764400 × 10⁻¹ r2 = 1.099477195358500 × 10² r3 = 1.662000000000000 × 10⁻² r4 = 8.994618670346399 × 10⁻⁵ r5 = 1.899634895912600 × 10⁻⁵ r6 = 8.077610866174112 × 10⁻⁵ r7 = 6.694928001321700 × 10⁻⁵

r1 = 4.539433460495040 × 10⁻¹ r2 = 1.099477558604532 × 10² r3 = 1.518314688928225 × 10⁻² r4 = 8.336386582323006 × 10⁻⁵ r5 = 1.745180668436959 × 10⁻⁵ r6 = 7.831410850410846 × 10⁻⁵ r7 = 6.629909782342582 × 10⁻⁵

Table 2. Coefficients in the impedance fitting function Figure 1. NASA Langley Flow Impedance Tube

(10)

(a)

frequency [Hz]

R,X

1000 2000 3000 4000 5000

-3 -2 -1 0 1 2 3 4 5

R, measured X, measured R, fitted X, fitted CT57 Liner

Extrapolated Region

frequency [Hz]

R,X

500 1000 1500 2000 2500 3000

-3 -2 -1 0 1 2 3 4 5 6

R, measured X, measured R, fitted [Ozyoruk]

X, fitted [Ozyoruk]

CT73 Liner

(b) (c) Figure 2. Use of grid points for impedance boundary condition

Figure 3. NASA ceramic tubular liners (a) Sketch (Ref. 10) and schematic of standing wave modes.

Measured impedance data and broadband fitting data for CT57 (b) and CT73 (c).

(11)

1000 Hz 2500 Hz 3000 Hz

2000 Hz 1500 Hz 500 Hz

x [inch]

SPL[dB]

0 8 16 24 32

80 90 100 110 120 130 140 150

Single Freq Model Broadband Model Impedances for CT73 liner,

fitted by Ozyoruk et al.

(1st order broadband model)

1000 Hz 2500 Hz 3000 Hz

2000 Hz 1500 Hz 500 Hz

x [inch]

SPL[dB]

0 8 16 24 32

80 90 100 110 120 130 140 150

Single Freq Model Broadband Model Impedances for CT73 liner,

fitted by Ozyoruk et al.

(2nd order broadband model)

(a) (b)

2000 Hz 1500 Hz 500 Hz

130 140

(c )

Figure 4. Comparison between single-frequency simulation and multi-frequency simulation (a) first- order approximation (b) second-order approximation in z-transform. (c) is the first-order approximation window-zoom of (a).

(12)

x [inch]

SPL[dB]

0 5 10 15 20 25 30

110 115 120 125 130 135 140 145

150 500 Hz

x [inch]

SPL[dB]

0 5 10 15 20 25 30

70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145

150 1000 Hz

x [inch]

SPL[dB]

0 5 10 15 20 25 30

110 115 120 125 130 135 140 145

150 1500 Hz

(a) (b) (c)

x [inch]

SPL[dB]

0 5 10 15 20 25 30

110 115 120 125 130 135 140 145

150 2000 Hz

x [inch]

SPL[dB]

0 5 10 15 20 25 30

110 115 120 125 130 135 140 145

150 2500 Hz

x [inch]

SPL[dB]

0 5 10 15 20 25 30

100 105 110 115 120 125 130 135 140 145

150 3000 Hz

(d) (e) (f)

Figure 5. SPL results from linear simulation of GIT; zero mean flow case (a) 500 Hz (b) 1000 Hz (c) 1500 Hz (d) 2000 Hz (e) 2500 Hz (f) 3000 Hz; Line : simulation, Circle : NASA experiment

(13)

(a)

(b)

Figure 6. Comparison of the simulation (red lines) and analytic solution (blue circles) for nonlinear plane wave propagation with fundamental frequency of 1000 Hz (a) Initially sinusoidal waves (left : 140 dB, right :

160 dB) (b) Initially saw-tooth waves (left : 140 dB, right : 160 dB)

overall

6th 2nd harmonic

3rd 4th 5th fundamental

SPL[dB]

0 20 40 60

135 140 145 150 155 160

165 Sawtooth solution

Simulation

frequendy = 1000 Hz (NPPW=202)

x [inch]

p[Pa]

0 20 40 60

-4000 -2000 0 2000 4000

x [inch]

p[Pa]

0 20 40 60

-4000 -2000 0 2000 4000

3rd 4th

5th fundamental

SPL[dB]

0 20 40 60

80 90 100 110 120 130 140 150 160

170 Fubini solution

Simulation

x [inch]

p[Pa]

0 20 40 60

-400 -200 0 200 400

overall

3rd 4th 5th fundamental

SPL[dB]

0 20 40 60

115 120 125 130 135 140

145 Sawtooth solution

Simulation

frequendy = 1000 Hz (NPPW=202) 2nd harmonic

4th fundamental

3rd

SPL[dB]

0 20 40 60

60 70 80 90 100 110 120 130 140

150 Fubini solution

Simulation

x [inch]

p[Pa]

0 20 40 60

-400 -200 0 200 400

(14)

overall

6th 2nd

3rd 5th 4th

fundamental

liner fundamental 2nd 3rd overall

x [inch]

SPL[dB]

0 10 20 30 40 50

50 60 70 80 90 100 110 120 130

fund. freq. = 1000 Hz OSPL = 120 dB (ref)

overall

6th 2nd

3rd 5th 4th

liner fundamental 3rd 2nd overall

fundamental

x [inch]

SPL[dB]

0 10 20 30 40 50

70 80 90 100 110 120 130 140 150

fund. freq. = 1000 Hz OSPL = 140 dB

(a) (b)

overall

6th 2nd

3rd 5th 4th

liner 2nd

fundamental overall

fundamental

x [inch]

SPL[dB]

0 10 20 30 40 50

90 100 110 120 130 140 150 160 170

fund freq = 1000 Hz OSPL = 160 dB

^{x [inch]}

SPL-SPLref[dB]

0 10 20 30 40 50

-70 -60 -50 -40 -30 -20 -10 0 10

3rd dotted : 120 dB line : 160 dB

2nd

fundamental fundamental

3rd 2nd

liner

(c) (d)

Figure 7. Nonlinear propagation of saw-tooth wave with fundamental frequency of 1000 Hz in CT73 lined duct (a) 120 dB source (b) 140 dB source (c) 160 dB source (d) Comparison between 120 dB and 160 dB

(15)

overall

liner 2nd : out of range overall

x [inch]

SPL[dB]

0 10 20 30 40 50

50 60 70 80 90 100 110 120 130

fund freq = 1000 Hz OSPL = 117.8 dB

overall

2nd liner

2nd fundamental overall

2nd (w/o liner)

(w/o liner)

x [inch]

SPL[dB]

0 10 20 30 40 50

70 80 90 100 110 120 130 140 150

(a) (b)

overall 2nd

3rd liner

2nd fundamental overall

fundamental

x [inch]

SPL[dB]

0 10 20 30 40 50

90 100 110 120 130 140 150 160 170

2nd (w/o liner)

fundamental (w/o liner)

3rd (w/o liner)

(c) (d)

Figure 8. Nonlinear propagation of sinusoidal wave with fundamental frequency of 1000 Hz in CT73 lined duct (a) 120 dB source (b) 140 dB source (c) 160 dB source (d) time history for 160 dB source

(16)

x [inch]

SPL-SPLref

0 5 10 15 20 25 30 35 40

-70 -60 -50 -40 -30 -20 -10 0 10

140 dB experiment 130 dB experiment

130 dB simulation (2nd) 130 dB simulation (fundamental)

140 dB simulation (2nd) 140 dB simulation (fundamental)

Figure 9. Comparison between nonlinear simulation and experimental data for 1 kHz source

Figure 10. Comparison between KAIST pediction (Solid Lines) and Boeing prediction (dashed line) of Nonlinear propagation of saw-tooth wave with fundamental frequency of 1000 Hz in CT57 lined duct