Time-Domain Impedance Formulation Suited for ...

13th AIAA/CEAS Aeroacoustics Conference (28th AIAA Aeroacoustics Conference)

AIAA 2007-3519

Time-Domain Impedance Formulation suited for Broadband Simulations Yves Reymen∗, Martine Baelmans†, and Wim Desmet.† Katholieke Universiteit Leuven, Department of Mechanical Engineering, Leuven, Belgium

In many aeroacoustic applications lining material plays an important role in controlling the emitted noise levels. To be able to study the effect of different materials and to judge their effectiveness at design time, adequate numerical models are essential. By means of numerical studies, optimization of the noise reduction becomes possible by sensitivity analysis of the various material parameters and the geometrical layout. Most of the time, the lining material is acoustically characterized in the frequencydomain. This is usually done by setting up a test with a single harmonic wave and then identifying the impedance from the measured response. The obtained complex value is the impedance at the frequency of the harmonic wave excitation. This procedure can only describe linear effects. Time-domain computational methods have a clear advantage over frequency-domain methods for broadband problems, non-linear interaction investigations and transient wave simulations. To be able to represent an impedance boundary in the time-domain, there is a need to ‘translate’ the frequency data to the time-domain. The proposed time-domain impedance formulation uses a 2-step approach: in the frequency-domain a sum of template functions is used to fit a set of impedance data. These template functions can be analytically transformed to the time-domain. With the assumption of piecewise-constant or piecewise-linear velocity, the convolution of the impedance model and the velocity can be efficiently performed by recursive convolution to give the pressure. A dual formulation for the admittance is obtained by switching the roles of the velocity and the pressure. To be physically feasible, a time-domain formulation has to comply with 3 necessary conditions. It has to be causal, real and passive. The general impedance/admittance model proposed by the authors is causal and real. A proper selection of parameters makes it also passive. The formulation is suited for broadband simulations by inclusion of additional template functions. An efficient implementation is obtained by performing recursive convolution. This technique requires only some additions and multiplications. The storage is limited to (complex-valued) accumulators in the boundary points. No time history of solution data is required, neither is there a need to compute time derivatives. This formulation is incorporated in a Quadrature-Free Discontinous Galerkin Method for the Linearized Euler Equations. This contribution considers the NASA Grazing Impedance Tube, a very well documented validation case that supplies experimental data for 26 frequencies at different mean flow speeds. The response of the lined duct for all frequencies is calculated in a single simulation.

I.

Introduction

Lining materials are frequently used in many aeroacoustic applications to reduce the emitted noise. To improve mechanical designs, good models are essential to study the effect of different materials. Simulation allows to optimize the noise reduction by varying the various material parameters and the geometrical layout. Typically, lining material is acoustically characterized in the frequency-domain. Data is gathered by setting up a test with a single harmonic wave and then identifying the impedance from the measured ∗ PhD

Student, Dept. of Mechanical Engineering, Celestijnenlaan 300, B-3001 Heverlee, Belgium, AIAA student member. Dept. of Mechanical Engineering, Celestijnenlaan 300, B-3001 Heverlee, Belgium.

† Professor,

1 of 11 American ofand Aeronautics and Copyright © 2007 by the author(s). Published by the American Institute ofInstitute Aeronautics Astronautics, Inc.,Astronautics with permission.

response. The obtained complex value is the impedance at the frequency of the harmonic wave excitation. This procedure, with the inherent assumption of a periodic input signal, can only describe linear effects. Time-domain computational methods have a clear advantage over frequency-domain methods for broadband problems, non-linear interaction investigations and transient wave simulations. To be able to represent an impedance boundary in the time-domain, there is a need to ‘translate’ the frequency data to the timedomain. At a given frequency ω, the pressure P (xb , ω) in the frequency-domain for a position xb on the lining material, is proportional to the normal velocity V (xb , ω) by the impedance Z(ω). A capital indicates the Fourier transform of a quantity. The equivalent expression in the time-domain involves the convolution of the inverse Fourier transform z(t) of the impedance with the velocity v(t).8 P (xb , ω) = Z(ω) · V (xb , ω) p(xb , t) = z(t) ∗ v(xb , t) =

(1) 1 2π

Z

∞

z(t − τ ) · v(xb , t)dτ

(2)

−∞

To be able to take the inverse Fourier transform of the impedance in equation (3), Z has to be known over the entire frequency range. This extension to all possible frequencies poses a first difficulty and establishes the need for an impedance model. Secondly the convolution has to be performed, which in its full form is a very costly operation (in terms of CPU time), and in addition requires the storage of the entire time history of the velocity. Z ∞

Z(ω)eiωt dω

z(t) =

(3)

−∞

For the impedance model to be physically feasible, it has to comply with 3 necessary conditions as indicated by Rienstra.8 It has to be causal, real and passive. These conditions are not inherently satisfied by a general polynomial fit to the frequency data. Fortunately, the second difficulty, the full convolution, can be avoided in many ways. These include, without going into further details, replacing iω by d/dt, the z-transform, and recursive convolution. Next, a non-exhaustive list is given of different approaches found in literature. Tam & Aurialt12 proposed a 3-parameter model, resembling a mass-spring-damper system, and replaced iω by d/dt. This leads to a formulation with very low computational cost, but it is not applicable as a general broadband model. Ozyoruk7 proposed a broadband impedance model based on a rational polynomial fit in combination with the z-transform. This model is rather sensitive to instabilities, but can be used as a general broadband model. Rienstra8 proposed a model based on a Helmholtz-resonator and the z-transform that satifies all conditions and can be exactly tuned to the impedance at a design frequency. Fung & Ju2 proposed a model for the reflection coefficient, relating incoming and outgoing velocities. This enables to apply a space-time continuation that allows for a non-causal model. The convolution is dealt with by recursion, an idea originally developed in the computational electromagnetics community. Recently the authors proposed a time-domain impedance formulation10 based on a 2-step approach: in the frequency-domain a sum of template functions is used to fit a set of impedance data. These template functions can be analytically transformed to the time-domain. With the assumption of piecewise-constant or piecewise-linear velocity, the convolution of the impedance model and the velocity can be efficiently performed by recursive convolution to give the pressure. A dual formulation for the admittance is obtained by switching the roles of the velocity and the pressure. The goal of this research is to demonstrate the potential of the formulation for general broadband simulations. The formulation is implemented within the framework of a Quadrature-Free Discontinous Galerkin Method for the Linearized Euler Equations9, 11 and applied to the NASA Grazing Impedance Tube,14 a very well documented validation case that supplies experimental data for 26 frequencies at different mean flow speeds. The response of the lined duct for all frequencies is calculated in a single simulation.

II.

Modelling of Impedance

Although the time-domain allows the description of non-linear effects, here the models are limited to linear interactions because we start from a frequency-domain impedance model. Looking back to equation (2), z(t) can be viewed in terms of linear systems theory as the unit impulse response of the system Z(ω). If the system can be described by a rational function, it can also be written 2 of 11 American Institute of Aeronautics and Astronautics

in the form of a partial fraction expansion with residues Ak and poles λk , equation (4), according to Fung & Ju.2 The poles have to be real λk or complex conjugated λl and λ∗l , given byα ± iβ. Each real pole determines a first-order system (5), each pair of complex conjugated poles determines a second-order system (7). Equations (6) and (8) give the corresponding unit impulse responses. The condition on λk and αl is to ensure causality. Z(ω) =

P X k=1

Ak iω + λk

(4)

Ak iω + λk zk (t) = Ak e−λk t H(t)

Zk (ω) =

(5) λk ≥ 0

Cl (iω) + Dl Bl Al = + iω + λl iω + λ∗l (iω + αl )2 + βl2 Dl − αl Cl −αl t zl (t) = e H(t) Cl cos(βl t) + sin(βl t) βl

(6) (7)

Zl (ω) =

αl ≥ 0

(8)

H(t) is the Heaviside function. II.A.

Impedance Model

A general impedance model can be written as the sum of S single pole systems and T complex conjugate pair systems. It can be easily shown that it is causal and real. For a good choice of the parameters it is also passive. T S X X Cl (iω) + Dl Ak + λk ≥ 0 & αl ≥ 0 (9) Z(ω) = iω + λk (iω + αl )2 + βl2 l=1

k=1

This model can effectively be applied to broadband simulations: the possibility to add extra terms allows to fit the model to a set of tabulated impedance values with any desired accuracy. For single-frequency simulations, a simplified model using only one pair of complex conjugate poles is sufficient.10 II.B.

Time-Domain Formulation

The convolution of the impedance impulse response z(t) and v(t), equation (2), can be calculated by recursive convolution thanks to the special form of zk (t) and zl (t) and the assumption that the velocity is piecewise constant or linear in a timestep ∆t. The derivation is inspired by the work of Luebbers6 on modelling of dispersive media in Computational Electromagnetics and is presented in previous work by the auhors.10 The resulting formulation of the time-domain impedance using recursive convolution and a piecewise linear approximation of the normal velocity is given by ψk (n∆t) =v(n∆t)

1 − e−λk ∆t e−λk ∆t (−λk ∆t − 1) + 1 + (v((n − 1)∆t) − v(n∆t)) + ψk ((n − 1)∆t)e−λk ∆t λk λ2k ∆t (10) (−αl +iβl )∆t

(−αl +iβl )∆t

1−e e ψˆl (n∆t) =v(n∆t) + (v((n − 1)∆t) − v(n∆t)) αl − iβl ˆ + ψl ((n − 1)∆t)e(−αl +iβl )∆t p(n∆t) =

S X k=1

Ak ψk (n∆t) +

T X l=1

Cl · Re{ψˆl (n∆t)} +

((−αl + iβl )∆t − 1) + 1 (−αl + iβl )2 ∆t

Dl − αl Cl · Im{ψˆl (n∆t)} βl

(11) (12)

where ψk and ψˆl are accumulators; the first is a real value, the second a complex value. In the formulation with the assumption of constant normal velocity within a time step, the second term is dropped in the first two equations. Here the formulation is written using the assumption of constant time step ∆t, but the formulation is equally valid for variable time steps: it suffices to replace n∆t by the time t.

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Only information from the previous time step is needed. The formulation can be applied within the stages of a Runge-Kutta time integration scheme that have varying step sizes. The formulation is given for the impedance, a similar formulation for the admittance is obtained by simply interchanging the role of pressure and velocity. If the order of the numerator is higher than that of the denominator in the rational function used to represent the impedance, using the admittance is necessary to be able to write the rational function as a partial fraction expansion (4). If the order of numerator and denominator are equal, both formulations can be used. In this case, an extra constant term appears in the expression that is trivial to incorporate. II.C.

Single Frequency Model

For the particular case of simulations containing only one single frequency, it is useful to have a specific model capable of matching any given impedance at that frequency. A suited frequency-domain model is the 3 parameter model, equation (13), proposed by Botteldooren13 and Tam & Auriault12 and also used by Ju & Fung.3 It can be considered as a mass-spring-damper model; all parameters Zi have to be positive to fulfill the 3 fundamental conditions.8 To be able to apply recursive convolution to this model, it is necessary to work with the admittance A, the inverse of the impedance. The model can then be written in the form of Zl . Equation (15) gives the relation between the parameter sets C, D, α, β and Z1 , Z0 , Z−1 . Z(ω) = Z−1 /(iω) + Z0 + Z1 (iω) = A(ω) =

(iω)2 Z1 + (iω)Z0 + Z−1 iω

(13)

C(iω) + D (iω + α)2 + β 2

C = 1/Z1

D=0

(14) α = Z0 /(2Z1 )

p β = Z−1 /Z1 − α2

(15)

At a design frequency ω ¯ , the impedance Z is given by the complex number R + iX. R is the resistance and X the reactance. Equation (16) gives the link between R, X and Z1 , Z0 , Z−1 from the 3-parameter model. The relation between R and Z0 is straightforward. X has to be somehow distributed over Z1 and Z−1 , while keeping the last two positive for a physically possible model. Here the choice is made to attribute a factor g of the absolute value of the reactance to one of the two Z-parameters and a matching value to the other depending on the sign of the reactance, see equation (17). The factor g has to be positive to make sure Z1 and Z−1 are positive. An additional condition on g follows from the expression for β: g has to be big enough p for β to be real. After some algebra the condition becomes g ≥ (−1 + 1 + (R/X)2 )/2. Z(¯ ω ) = R + iX = Z0 + i (Z1 ω ¯ − Z−1 /¯ ω) Z0 = R,

II.D.

(1 + g)|X| and Z−1 = g|X|¯ ω if X > 0 ω ¯ g|X| Z1 = and Z−1 = (1 + g)|X|¯ ω if X < 0 ω ¯

Z1 =

(16) or (17)

Properties

The formulation exhibits following properties: Efficiency: The formulation requires just a few additions and multiplications per time step. Low storage: Storage is needed only for the accumulators in the boundary points. No time history of solution data is required. Easy implementation: There are no (high-order) time derivatives in the formulation, which allows it to be incorporated in all discretization schemes. Variable time steps are easily accommodated. True broadband formulation: The model gives sufficient freedom to approximate any set of sampled impedance values.

III.

Implementation

The impedance and admittance formulations are implemented in the framework of a Quadrature-Free Discontinuous Galerkin Method for the Linearized Euler Equations,9, 11 supplemented with a low-storage Runge-Kutta time integration.1 This method allows to simulate acoustic propagation through (non-)uniform mean flows for 3D geometries and properly account for reflection, refraction and convection effects. 4 of 11 American Institute of Aeronautics and Astronautics

IV.

NASA Grazing Incidence Tube

The NASA Grazing Incidence Tube,14 see also figure 1, is used to characterize different types of liners. In the experiment, plane waves (0.5 to 3kHz) are generated at the inlet, that propagate through the hard-walled duct. In the middle section a piece of liner to be tested is installed. The mean flow through the duct can be varied and tests are performed with Mach numbers ranging form 0 to 0.5 on the centerline.

Figure 1: Measurement setup used for the NASA Grazing Incidence Tube14

IV.A.

Fitted impedance models

For the single-frequency models, the procedure, outlined in section II.C, can be followed using the experimental data for R and X at the specific frequency. The single frequency models match these data exactly. For the broadband models, a continuous approximation to the admittance/impedance is automatically fitted through the complex data using a procedure called Vector Fitting.15 This method uses an iterative approach to fit P poles λk and P residuals Ak as defined in equation 4. The poles are garanteed to be real and causal. These poles and residues can be easily converted to the parameters in the general broadband model of equation 9. A Matlab routine can be downloaded from the website of the author.16 If the fit is not passive from the first try, algorithms can be applied to obtain a passive model by slightly modifying the parameters. Figure 2 shows the original data and the vector fit with 10 poles of the admittance for the case without flow and with a 130dB plane wave excitation. The procedure resulted in 2 real poles and 4 pairs of complex conjugated poles. This means that S = 2 and T = 4 in equation 9. The amplitude is fitted with on average 5 percent accuracy. Figure 3 shows the bode plot of the fit. The phase, plotted in the bottom figure, shows that for all frequencies, the phase stays within the interval [−90 ◦ , 90 ◦ ]. In other words, the real part remains positive; the model is passive. Approximation of admittance

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IV.B.

Results

In the simulation, the inlet section is modeled as a damping zone in which a sum of 26 plane waves, with frequencies from 500 to 3000 in steps of 100Hz, is introduced. They all have equal amplitude and identical phase. The hard wall is modeled as a perfectly rigid wall with full reflection; the liner is modeled using the proposed time-domain impedance formulation. The outlet section is assumed to be anechoic and is implemented as a damping zone, which ensures low reflections at the outlet. The mean flow is modeled as a parabolic function with the correct average mean flow speed. This shear flow approximation seems to be a reasonably good approximation of the real mean flow profile.3, 4 As the mean flow is zero near the wall for a parabolic profile, no correction for the grazing flow needs to be applied to the impedance values. The domain is discretized with hexahedral elements with size h 1 inch (0.0254 meter) in each direction. The duct is 32 inches long and 2 inches high. In the third direction, one element is used and periodic boundary conditions are imposed to simulate in a quasi two-dimensional way. In the inlet and outlet zone, there are respectively 10 and 5 elements in longitudinal direction. In total, the mesh contains 47×2×1 elements. The polynomial order p in the Discontinuous Galerkin formulation is 2; this gives a spatially third order accurate method. According to the (2 ∗ p + 1) ' κkh rule proposed by Ainsworth,17 with k the wavenumber and κ a constant equal to about 3, this mesh is valid for frequencies upto 3600Hz. During each simulation a time series of pressure data is recorded on the wall opposite of the liner section. This time series is processed with a Fast Fourier Transform(FFT) to obtain results in the frequency-domain. The simulations use a time step of 4 · 10−6 seconds. The single frequency simulations are run from 0 to 0.014 seconds and the data from the last 0.01 seconds are used for the FFT. The multiple frequency simulation needs a higher frequency resolution and is run sfrom 0 to 0.1 seconds. Here data from the last 0.04 seconds is used for the FFT. Figure 4 compares the measurements data with the numerical results. The red circles represent the measurements. The green plus-signs represent within each subfigure the results obtained from a simulation in which only one plane wave with that particular frequency is introduced; such a simulation is named single frequency. The blue solid line represents the results from a simulation in which multiple planes waves for each of the 26 frequencies are introduced; this simulation is named multiple frequency. For all frequencies, except 500Hz, the results from the single frequency simulations correspond very well to the measured data. At 500Hz, the assumption of anechoic termination in the measurements was questionable and may have corrupted the measurements.14 The results from the multiple frequency simulation are quite 6 of 11 American Institute of Aeronautics and Astronautics

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Figure 4: Sound Pressure Level along upper wall for Mach number 0 as a function of the distance along the duct in inches (blue −: multifrequency simulation, green +: single frequency simulation, red o: reference data from measurements


good for 1000Hz, 1500Hz and 2500Hz; for the other frequencies substantial deviations are visible. Possible reasons for the unfavorable comparison to the measurements could be an interaction between the plane waves at different frequencies or a high sensitivity to the accuracy of the fit, to the quality of the frequency-domain model. To test the first reason, single frequency simulations are performed with the broadband model for the liner. Results are shown in figure 5. The results from the single frequency and the multiple frequency model almost coincide for every frequency. From these results we can conclude that there is no significant influence from one plane wave on another, as aspected for a linear algorithm.

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Figure 5: Sound Pressure Level along upper wall for Mach number 0 as a function of the distance along the duct in inches (blue −: multifrequency simulation, green +: single frequency simulation with broadband admittance model, red o: reference data from measurements) To test the sensitivity of the results to the accuracy of the model, single frequency simulations are performed at 2500Hz with the exact impedance data, with a 5% increase in the magnitude of the impedance and a 5% decrease in magnitude. These percentages are representative for the accuracy of the fit, shown in figure 2. Figure 6 compares the results from these 3 simulations. The small variations of the impedance data lead to large variations in the pressure at the end of the duct: there is a 5dB difference which translates to almost 80 percent difference in pressure as a result of 10 percent difference in impedance amplitude. This demonstrates the high sensitivity of the results to the quality of the fit for the admittance impedance. For the simulations with flow, center line Mach number of 0.3, the representation of the mean flow is an additional source of error. Here the choice is made to represent the flow by a parabole that vanishes at the wall, to avoid the need for corrections due to the grazing flow. The measured flow has a profile that looks more like a parabole superimposed on a constant mean flow.4, 14 Figure 7 shows the results. The single frequency results have become a little worse, in part because of the representation of the mean flow. The multiple frequency result are far off: the accumulation of errors has become very large.



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Figure 6: Sound Pressure Level along upper wall for Mach number 0 and single frequency simulations at 2500Hz as a function of the distance along the duct in inches (blue −: exact measurement data, green 4: +5%, red 5: −5%)

V.

Conclusions and future work

To be physically feasible, a time-domain formulation has to comply with 3 necessary conditions. It has to be causal, real and passive. The general impedance/admittance model10 proposed by the authors is causal and real. A proper selection of parameters makes it also passive. The formulation is suited for broadband simulations by inclusion of multiple poles. An efficient implementation is obtained by performing recursive convolution. This technique requires only some additions and multiplications. The storage is limited to (complex-valued) accumulators in the boundary points. No time history of solution data is required, neither is there a need to compute time derivatives. The broadband formulation has been applied to the NASA Grazing Incidence Tube benchmark and allows to characterize the broadband response of a liner in a single simulation. It was shown that the results are very sensitive to the quality of the fit used to approximate the impedance in this kind of simulations. So far, the mean flow has been assumed to be parabolic. Recent work5 indicates more accurate representations improve the agreement with experimental data. Also the assumption of an anechoic outlet is questionable at low frequencies. Both these topics need to be further investigated.

Acknowledgments The authors thank Michael G. Jones for providing the database of measurements of the NASA Grazing Incidence Tube. The research of Yves Reymen is supported by a fellowship of the Institute for the Promotion of Innovation through Science and Technology in Flanders (IWT-Vlaanderen).

References 1 Carpenter,

M.H. , Kennedy, C.A., “A fourth-order 2N-storage Runge-Kutta scheme”,NASA TM 109112, June 1994. K.-Y., Ju, H., “Broadband Time-Domain Impedance Models”, AIAA Journal, Vol. 39, No. 8, 2001, pp. 1449-1454. 3 Ju, H., Fung, K.-Y., “Time-Domain Impedance Boundary Conditions with Mean Flow Effects”, Vol. 39, No. 9, 2001, pp. 1683-1690. 4 Li, X. L., Thiele, F., “Numerical Computation of Sound Propagation in Lined Ducts by Time-Domain Impedance Boundary Conditions”, 10th AIAA/CEAS Aeroacoustics Conference, AIAA paper 2004-2902. 5 Li, X., et al., “Time-domain impedance boundary conditions for surfaces with subsonic mean flow”, Journal of the Acoustical Society of America, Vol. 119, No. 5, 2006, pp. 2665-2676. 2 Fung,


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Figure 7: Sound Pressure Level along upper wall for Mach number 0.3 as a function of the distance along the duct in inches (blue −: multifrequency simulation, green +: single frequency simulation with broadband admittance model, red o: reference data from measurements)


6 Luebbers, R. J., “FDTD for Nth order Dispersive Media”, IEEE Transactions on antennas and propagation, Vol. 40, No. 11, 1992, pp. 1297-1301. 7 Ozyoruk, Y., Long, L.N., Jones, M.G., “Time-Domain Numerical Simulation of a Flow-Impedance Tube”, Journal of Computational Physics, Vol. 146, 1998, pp. 29-57. 8 Rienstra, S. W., “Impedance models in time domain”, Messiaen - Project AST3-CT-2003-502938, Deliverable 3.5.1 of Task 3.5. 9 Reymen, Y., et al., “A 3D Discontinuous Galerkin Method for Aeroacoustic Propagation”, Twelfth International Congress on Sound and Vibration, 2005, paper 387. 10 Reymen,Y., et al., “Time-Domain Impedance Formulation based on Recursive Convolution”, 12th AIAA/CEAS Aeroacoustics Conference, AIAA paper 2006-2685. 11 Reymen, Y., et al., “Time-domain simulation of 3D lined ducts with flow by a unstructured Discontinuous Galerkin Method”, Fourteenth International Congress on Sound and Vibration, 2007, paper 519. 12 Tam, CKW, Auriault, L., “Time-Domain Impedance Boundary Conditions for Computational Aeroacoustics”, AIAA Journal, Vol. 34, No. 5, 1996, pp. 917-923. 13 Botteldooren, D., “Finite-diference time-domain simulation of low-frequency room acoustic problems”. Journal of the Acoustical Society of America, Vol. 98, No. 6, 1995. 14 Watson, W., et al., “Benchmark Data for Evaluation of Aeroacoustic Propagation Codes with Grazing Flow”, 11th AIAA/CEAS Aeroacoustics Conference, AIAA paper 2005-2853. 15 Gustavsen, B., Semlyen, A., “Rational approximation of frequency domain responses by vector fitting”, IEEE Transactions on Power Delivery, Vol. 14, No.3, 1999, pp. 1052-1061. 16 The Vector Fitting website. http://www.energy.sintef.no/Produkt/VECTFIT/index.asp . 17 Ainsworth, M., “Dispersive and dissipative behaviour of high order discontinuous Galerkin finite element methods”, Journal of Computational Physics, Vol. 198, pp. 106-130, 2004.
