Simulation of Acoustic Fields in Resonator–Type Problems Using Unstructured Meshes

Tatiana K. Kozubskaya*, Ilya V. Abalakin†, and Andrey V. Gorobets‡

Institute for Mathematical Modelling RAS, Moscow, Russia, 125047

and

Alexey K. Mirovov§

Central Institute of Aviation Motors, Moscow, Russia, 111116

The paper is devoted to numerical simulation of model resonator-type problems presenting a typical problem of nonlinear aeroacoustics. An interest to the problems of this kind is caused by the refined algorithms needed for their adequate prediction as well as by their practical importance for engineering applications. The numerical algorithm in use is based on the high-accuracy mixed FV/FE vertex-centered scheme for unstructured meshes. Three basic problems are considered: resonator chamber (i), impedance tube (ii), channel with resonators installed into the walls (iii). The model problems under study underlie the modeling of acoustic liners.

I. Introduction HE resonator–type problems can be considered as a typical problem in nonlinear aeroacoustics. One of the actual examples of such problems is simulation of noise suppression in resonant liners. An effective design of

noise absorbing panels needs a deeper understanding of determining physical mechanisms, from one hand side, and a convenient testing simulator for their quick optimization, from another. A wide range of accessible geometrical parameters, possible amplitude and frequency characteristics of incoming acoustic signal complicate the experimental study, and make the numerical simulation an attractive testing tool.

T

At simulating the resonator–type problems numerically, specifics are that in the case of high acoustic power of incoming signal, the computational domain under study covers both “linear” and “nonlinear” regions. The mathematical models in use have necessarily to take account of nonlinear effects. In the “linear” region, the acoustic wave propagation being a determining physical process needs high accuracy numerical algorithms for its proper resolution. In “nonlinear” regions, the corresponding numerical techniques have to be adaptable to the high gradients of solution up to the possible discontinuities. That is why the algorithms of nonlinear aeroacoustics have to combine the better properties of linear CAA and nonlinear CFD adaptively within a universal method applicable to both cases since normally there is no permanent space bound between “linear” and “nonlinear” operating zones.

The papers of C.K.W.Tam et al1,2 develop a universal algorithm of that kind basing on the DRP scheme well applicable to Cartesian grids. In the case of arbitrary unstructured meshes, a difficulty in constructing the schemes becomes stronger due to the necessity to provide a high accuracy of approximation on irregular space stencils. The matters are somewhat improved by the assumption on a sufficient smoothness of the solutions under consideration allowing only for weak jumps.

The paper presents a mixed finite–volume (FV) finite–element (FE) algorithm for solving nonlinear aeroacoustics problems on unstructured meshes. It is based on the high accuracy vertex–centered multi–parameter scheme3,4,5. It is built in a way of superconvergence so that it reaches its up to the 6th theoretical order of accuracy on the Cartesian meshes. A set of 1D and 2D test cases performed confirm the scheme properties.

Different formulations of resonator–type problems are considered. The first simulates a resonator chamber of a cylindrical shape. That formulation serves also for the verification of numerical techniques used by comparing the * Dr., Senior Researcher, Head of CAA Laboratory, 4-A, Miusskaya Sq., AIAA Member. † Senior Researcher, CAA Laboratory, 4-A, Miusskaya Sq. ‡ Junior Researcher, CAA Laboratory, 4-A, Miusskaya Sq. § Dr., Senior Researcher, 2, Aviamotornaya St.

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12th AIAA/CEAS Aeroacoustics Conference (27th AIAA Aeroacoustics Conference)8 - 10 May 2006, Cambridge, Massachusetts

AIAA 2006-2519

Copyright © 2006 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

results with the a priori known theoretical or experimental data on the resonator properties. The second group of problems presents numerical experiments in the impedance tube ended by a resonator box which is separated by a perforated screen. In the both cases the flow detected inside resonators is originated only by incoming acoustic waves. The third problem under study considers a subsonic flow in the channel with walls included built-in resonators. That formulation is close to the conditions of typical physical experiments on designing acoustic liners. A computational goal of the present study is to verify an applicability and robustness of the proposed algorithm on specific resonator–type problems while a physical one is to glance inside resonators, in order to approach to better understanding of gas dynamics mechanisms underlying the efficiency of noise absorption in acoustic liners.

II. Problem Formulation Three basic mathematical models, namely the Navier–Stokes equations (i), the Euler equations (ii), and the

linearized Euler equations (iii), are used in the present study. The formulations in plane and axisymetric geometry are considered. The following general hyperbolic–type form of writing all the above–listed 2D models is adopted:

in the case of plane geometry 1 .

Re

xy xyxy NS NS

t x y x y⎛ ⎞∂ ∂∂ ∂ ∂

+ + = + +⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠

F GQ F G H

in the case of axisymmetric geometry ( ) ( ) ( ) 1 .

Re

rz rzrz NS NSr r r

t z r z r∂ ∂ ∂ ⎛ ⎞∂ ∂

+ + = + +⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠

Q F G F GH

III. Numerical Algorithm The numerical scheme considered is based on the semi–discrete approximation within the mixed FV/FE

approach 1

i

ijji

ddt C ∈Ω

+ Φ =∑Q H

where , are the fluxes through the edges of cell surrounding the node i. , ij ijΦ ∈Ω

High order flux is determined as ( )HO CD, , , ,E E E E

ij i j i j ij ij ij= −Φ F F G G n Φ D

( ) ( )CD 1 ,2

E x E y E x E yij ij ij ij ijj i

n n n n⎡ ⎤= + + +⎢ ⎥⎣ ⎦Φ F G F G

( ) ( )1 sign2 2

E x E y E x E yi iij ij ij ij ijj i

n n n n+⎛ ⎞ ⎡ ⎤= δ + − +⎜ ⎟ ⎢ ⎥⎣ ⎦⎝ ⎠

Q QD A F G F G

( ) ( ) 1sign diag sign ,sign ,sign ,signn n n nu u u c u c −= +A S S−

where is a central–difference flux and is a dissipative flux determined within the non–MUSCL method as it is proposed in papers

CDijΦ ijD

6. The fluxes are calculated as functions on the extrapolated values EF and EG at the interface between the cells

( ) ( )1 1,2 2

E Ei i j jij ji= − ∇ ⋅ = + ∇ ⋅F F F ij F F F ij

where the extrapolation slopes are defined with the help of gradients of different approximation types. An accuracy of defining the extrapolating slopes determines the scheme accuracy in total. The details of the scheme construction are given in paper3,7

It should be noted that according to the supercovergence property of the scheme, the highest theoretical accuracy order, up to the 6th order depending on the scheme parameters, is reached on the structured Cartesian meshes. In this connection it is preferable if possible to cover the computational domain with the “Cartesian” elements and use the unstructured mesh for filling the space in the vicinity of curvilinear shapes. In doing so, it is clear that the smaller size of Cartesian mesh corresponds to the narrower unstructured edging.

In order to make the algorithm universal that is applicable for linear and nonlinear problems at once, let us introduce an adaptive dissipation in a form of smooth switching from the high order flux to the lower order approximation as follows

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( ) ( )(1 ) , , , , , , , ,HO E E E E LO lim lim lim limij ij i j i j ij ij i j i j ij= −α + αΦ Φ F F G G n Φ F F G G n .

Here are the extrapolated values at the interface between the cells and are calculated with the use of the limited extrapolation slopes.

, , ,lim lim lim limi j i jF F G G

The algorithm is to be completed with accurate boundary conditions at free bounds, in order not to prevent from getting the high accuracy solution inside the domain. In the paper two types of boundary conditions are used: non–reflecting radiation and outflow conditions8 and the conditions based on the Steger–Warming flux splitting9.

IV. Numerical Results

A. Testing The numerical scheme in use is verified on a set of 1D and 2D benchmark problems. A brief overview of the

results on the two of them is given below. 1. 1D Benchmark problem

The first test case is a problem of the Fourth CAA Workshop on Benchmark problems10. It concludes in solving the 1D convective wave equation

0u ut x

∂ ∂+ =

∂ ∂

with an initial wave profile

[ ] 2( ,0) 2 cos(1.7 ) exp ln 2( /10)u x x x⎡ ⎤= + −⎣ ⎦ . The numerical results of applying the 6th

order scheme are presented graphically in Fig.1 and in the form of tables showing the error and

numerical scheme order correspondingly in Fig. 2. satisfactory while starting from the step ¼ and smalexpected in theory.

2. 2D Benchmark problem The problems 1 and 2 of Category 3 of the Sec

2D test case. Both problems are governed by the linea

1L norm 2L norm L ∞ no1x∆ = 5.35×101 3.46 8.29×11 2x∆ = 2.78×101 4.88 1.49

1 4x∆ = 8.37×10-1 1.68×10-1 5.99×1

1 8x∆ = 1.35×10-2 2.70×10-3 9.72×1

1 16x∆ = 2.12×10-4 4.25×10-5 1.53×1

Figure 2. Tables of computational errornorms versus the mesh step.

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Figure 1. Comparison of the numerical solution obtainedby the 6th order scheme with the exact solution at time=80.

The results obtained with the mesh steps 1 and ½ are not ler the scheme demonstrates its accuracy up to the 6th order

ond CAA Workshop on Benchmark problems11 are taken as a rized Euler equations and correspond to the linear propagation

rm0-1

0-2

0-4

0-5

1L norm 2L norm L ∞ norm1x∆ =

9.42×10-1 -4.95×10-1 -8.48×10-11 2x∆ = 1 2x∆ =

5.06 4.86 4.64 1 4x∆ = 1 4x∆ =

5.96 5.96 5.94 1 8x∆ = 1 8x∆ =

5.99 5.99 5.99 1 16x∆ =

s (left) and numerical scheme orders (right) in different

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of the Gaussian pulse in the subsonic (M=0.5) background flow in the horizontal and diagonal direction with respect to the computational domain coordinates correspondingly.

The table in Fig. 3 gives the values of computational errors and accuracy norms obtained at the predictions using the median (MC) and circumcenter (CC) cells. Fig.4–5 demonstrate the centerline cuts of density perturbation predicted when using different types of meshes (Cartesian or unstructured triangular), number of nodes, and scheme accuracy (5th or 6th).

Structured mesh

Diagonal backgroundflow

Unstructured mesh Horizontal

background flow

10201 40401 9693 39527

MC-5 Nord

9.91×10-1

2.39 1.89×10-1 1.02×100

1.99 1.41×10-1

MC-6 Nord

1.20×100

3.20 1.31×10-1 1.25×100

2.54 9.88×10-2

CC-5 Nord

7.93×10-1

2.93 1.04×10-1 1.01×100

1.97 1.42×10-1

CC-6 Nord

8.69×10-1

4.05 5.22×10-2 1.29×100

1.98 1.80×10-1

Figure 3. Computational errors and accuracy orders in L2 norm.

F(

Figure 4. Density perturbation, “diagonal” test. Cartesian meshes 101x101 (left) and 201x201 (right),time 40, 6th order of accuracy.

igure 5. Density perturbation, “horizontal–wind” test. Unstructured meshes of 9693 (left) and 39527

right) nodes, time 40, 5th order of accuracy.

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3. 2D Riemann problem Let us consider the 2D Riemann problem with a

weak discontinuity representing by the following initial data:

2 0( , ,0)

1 0

0 0( , ,0)

1 0

2 0( , ,0)

1 0

xx y

x

xu x y

x

xp x y

x

<⎧ρ = ⎨ >⎩

<⎧= ⎨ >⎩

<⎧= ⎨ >⎩

Fig. 6 shows the stabilizing impact of local artificial dissipation applied to the 5th order scheme

Figure 6. Density profile at time=40: exact solution (black), numerical solution obtain by the 5th order scheme (blue), and by the 5th

order scheme with artificial dissipation (red)

B. Resonator Chamber The first problem under consideration is an

excitation of flow inside a resonator chamber under the action of incoming acoustic wave. The chamber under study represents an axisymmetric cylindrical–shape resonator schematically shown in Fig. 7. Monochromatic acoustic waves of different frequencies and power enter the chamber from the left. In contrast with a classical Helmholtz resonator, a configuration with rather wide and short resonator throat is studied. The problem is simulated by the Navier–Strokes equations in the axisymmetric geometry. The cases of frequencies 150, 195 and 230 Hz with the power of 130 and 160 Db are considered.

It is found that the quasi-periodic oscillations settled inside a chamber sufficiently depend on a frequency of the incoming wave and its power. In the case of 130 Db, the amplitude of incoming wave is very low and the results well coincide with the ones obtained by the linearized models. It can be demonstrated by comparing the spectra of pressure pulsation in the resonator throat. Fig. 8 demonstrates that in the case of 160 Db–incoming wave the spectrum of the pressure pulsation in the resonator center point has two well seen overtones of incoming harmonics.

Fig 9 shows he mean flow fields inside the resonator in the cases of frequencies 150, 195 and 230 Db. The mean flow fields are calculated by time averaging over a large number of wave periods. As it is seen a system of vortices is formed in the vicinity of resonator internal corner.

Figure 7. Scheme of computational domain with dimensions(mm)

Figure 8. SPL: Difference in spectra for low and high power signals monochromatic (195 Hz) waves

Figure 9 Mean flow fields in resonator chambers at frequencies 150 Hz (left), 195 Hz (middle), 230 Hz (right)

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C. Impedance Tube The second problem considered is

devoted to the simulation of acoustic and flow phenomena in the impedance tube ended by a resonator chamber of length equal to a quarter of incoming wave length. The computational domain represents a long tube, a little longer than an incoming wave length, the resonator is separated from the tube by a perforated screen with a narrow orifice connecting the resonator with the rest tube region.

Fig. 10, A shows schematically the computational domain and its decomposition into 40 subdomains used at parallel computing on multi–processor computer system. The decomposition demonstrates the corresponding mesh refinement in the vicinity of resonator throat.

The problem is governed by the Navier-Stokes equations that is why the mesh is condensed to the walls to provide a better resolution of boundary layers (see Fig . 10, B, C).

Fig. 11 demonstrates a significant difference between the results produced by the linearized Euler equations, Euler equations and the Navier–Stokes equations. The most realistic results well corresponding to the experimental data are obtained by the Navier–Stokes equations. The numerical study of the acoustic and flow fields inside the imdedance tude is carried out basing on that mathematical model in the plane geometry.

The main problem parameters used at the predictions are the following: incoming acoustic wave of frequency 273 Hz and power 147.1 Db; geometrical parameters: diameter of the tube is 2.35 cm, diameter of the orifice is 0.75 cm, the width of separated screen is 0.6 cm. Under the above conditions it is found that in the vicinity of the resonator throat the incoming purely acoustic wave loses its energy by transforming into the flow, generation of eddies as well as dissipated due to the viscosity and heat conductivity. The formation of pairing vortices is around the orifice shown in Fig. 12

Figure 10. Impedance tube: scheme of the domain together with the decomposition into 40 subdomains for parallel computing (top); unstructured triangular mesh fragments (bottom).

Figure 11. The difference between the results obtained by using the linearized Euler equations (top, left), Euler equations (top and bottom, right), and the Navier–Stokes equations (bottom, left)

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Figure 12. Generation of pairing vortices andacoustic flow in the vicinity of resonator throat

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D. Channel with Resonators Installed into Walls The third second problem under study is the closest to the conditions of physical experiment on acoustic resonant

liners. It simulates numerically the subsonic flow together with acoustic radiation in the channel either with a resonator or a system of resonators installed into the walls.

In the present study the following conditions are used: the incoming acoustic wave of frequency 3431 Hz and power 150 Db accompanies the upstream subsonic flow from the left at Mach number 0.4 (approximately corresponding to the engine regime at aircraft landing). The first predictions are carried out for the case when only one resonator is built into the channel wall. The scheme of computational domain together with a variant of domain decomposition for 40 processor nodes is demonstrated in Fig. 13. Fig 13 also gives the problem dimensions in mm.

The problems of that type as all the resonator–type problems present a serious difficulty connected with a great difference in scales, for instance, between the channel diameter and the diameter of orifice equal to 2 mm. The fragment of grid used at the predictions are drawn in Fig. 14.

The problem is solved using both the Euler and Navier–Stokes equations. When using the Euler equations one can detect a well noticeable whistle caused by the mixing layer development between the orifice edges (see Fig. 15) and multiple reflections inside the resonator throat. In contrast, when using the Navier–Stokes equations the low speed flow in the vicinity of wall (corresponding to rather thick

Figure 13. Channel with resonators installed into walls: scheme of the computational domain with dimensions (mm) and main control points (top); decomposition into 40 subdomains for parallel computing (bottom)

Figure 14. Fragments of triangular meshes used at the predictions

boundary layer in the channel) cardinally changes the flow dynamics in the resonator throat. For instance, no whistle is found. It is well seen in Fig. 16 where a comparison of the results obtained by using the Euler and Navier–Stokes systems is shown at different stages of the process. The right picture in Fig. 15 also presents a field of turbulence kinetic energy calculated by the

F

igure 15. Development of mixing layer between the edges of orifice (left and middle); turbulence kinetic energy

(right)

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time averaging in the process of computation. As it is expected the maximum of turbulence kinetic energy is located along the centerline of the mixing layer. Fig 17 shows the distortion of incoming long wave acoustic radiation due to the high frequency cylindrical waves (whistle) produced by the flow interaction with the orifice under the Euler

p

formcases

Figure 16 Shade pictures of density disturbances corresponding to experimental method of Schlierenhotography. Initial and developed stages predicted by the Euler equations (top pictures), and by the

Navier–Stokes equations (bottom pictures)

ulation. The same Fig 17 also demonstrates a difference in pressure pulsation spectra in the viscous and inviscid in the point (0 mm, 10 mm) (see Fig. 13).

Fb

igure 17. Incoming wave distortion due to the high frequency cylindrical waves radiating y the orifice

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The case of a system of five resonators installed into the channel wall is considered under the Euler formulation. The corresponding flow patterns with the visible acoustic fields are presented in Fig. 18

Figure 18. Initial and developed stages in the case of five resonators installed into the channel wall.

V. Conclusion The work carried out can be considered as a step towards facing the refined numerical techniques developed by

the authors to the engineering applications in aeroacoustics. The results achieved on this way are still mostly of qualitative nature, a good correlation with the experimental data in quality and not in quantity yet is found. It is mainly connected with a model character of the predictions and a lack of corresponding experimental data available. At the same time, the predictions performed demonstrate an efficiency of adaptation of high accuracy numerical algorithms to the unstructured meshes and promising possibilities of this way for solving practical engineering problems.

When considering the resonator–problems, one can find also a difficulty in the numerical data treatment, in order to extract the meaningful technical characteristics. For instance, it is connected with the necessity to distinguish the incoming and outcoming acoustic waves, or to extract the acoustic energy from the energy of total flow, and so on. Although the problems of that kind are under consideration of many authors in aeroacoustics (see, for example, paper1) they seemingly remain not solved yet in full measure and invites for further development of the techniques for the numerical data processing on the way not of reproducing numerically the methods used at physical experiments, but of elaborating the original methods by taking the advantages of numerical simulation.

Acknowledgments The work presented was supported by the Russian Foundation for Basic Research (Grants No 04-01-08034, 05-

07-90230, 06-01-0293). The authors thank Prof. Alain Dervieux (INRIA Sophia–Antipolis, France) for the long term collaboration in

developing the numerical algorithms used in the predictions, Dr. Irina Lebedeva for the formulation of impedance-tube problems and also appreciate the help of Vladimir Bobkov, junior researcher at IMM RAS, and Mikhail Chernetsov, Ph.D. student at Moscow Physics and Technology Institute, in the generation of grids.

References 1Tam C.K.W., and Ju H., “A computational and experimental study of slit resonators”, AIAA paper 2003-3310, 2003. 2Tam C.K.W., and Shen H., “Direct computational of nonlinear acoustic pulses using high order finite difference schemes”,

AIAA paper 93-4325, 1993. 3Abalakin, I.V., Dervieux, A., and Kozubskaya T.K., “A vertex centered high order MUSCL scheme applying to linearised

Euler acoustics”, INRIA report RR4459, April 2002. 4Debiez, C., and Dervieux, A., “Mixed element volume MUSCL methods with weak viscosity for steady and unsteady flow

calculation”, Computer and Fluids, Vol. 29, 1999, pp. 89-118. 5Gourvitch N., Rogé G., Abalakin I., Dervieux A., and Kozubskaya T., “A tetrahedral–based superconvergent scheme for

aeroacoustics”, INRIA report RR5212, May 2004.

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6Huang L.C., “Pseudo-Unsteady Difference Schemes for Discontinuous Solution of Steady-State, One-Dimensional Fluid Dynamics Problems”, Journal of Computational Physics, Vol. 42, 1981, pp. 195-211.

7Abalakin, I., Dervieux, A., and Kozubskaya T., “Computational Study of Mathematical Models for Noise DNS”, AIAA Paper 2002-2585, 2002.

8Tam C.K.W., and Webb J. C., “Dispersion-Relation-Preserving Schemes for Computational Aeroacoustics”, Journal of Computational Physics, Vol. 107, 1993, pp. 262-281.

9Hirsh C., Numerical computation of internal and external flows, Vol. 2, John Wiley & Sons, 1998. 10Fourth Computational Aeroacoustics (CAA) Workshop on Benchmark Problems [online], Cleveland, Ohio, October 20-22,

2003, URL: http://www.math.fsu.edu/CAA4/pdfs/Category1/problem1.pdf. 11Proceedings of ICASE/LaRC Workshop on Benchmark Problems in Computational Aeroacoustics (CAA), Hampton,

Virginia, October 24-26, 1994, Edited by J.C.Hardin, J.R.Ristorcelli, and C.K.W.Tam, May, 1995.

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