Aeroacoustic Noise Source Mechanisms in Simple

Expansion Chambers

G. Rubio∗, W. De Roeck, J. Meyers, M. Baelmans, and W. Desmet

K.U.Leuven, Department of Mechanical Engineering, Leuven, Belgium

Expansion chambers are often installed in flow duct systems to attenuate the noisegenerated by combustion engines, fans,... However, in certain circumstances, they canbecome flow-excited noise generators rather than silencers. In the present paper, a LargeEddy Simulation approach is used to predict the flow-induced noise in a simple 3D expan-sion chamber, in order to gain more insight in the physics of the flow-acoustic interactionresponsible for the tonal noise components such as tail pipe and chamber resonances.

Nomenclature

A = Van Driest dissipation constantCρ = Dissipation constant for the continuity equationCs = Smagorinsky constantcp = Specific heat at constant pressurec = Speed of soundD = Height of the inlet pipee = Total energy per mass unitf = Frequencyk = Ratio of the convection velocity of the vortices to the free-stream velocityM = Mach numbern = Mode numberL = Chamber lengthLc = Characteristic acoustic lengthLt = Tail pipe lengthPrt = Turbulent Prandtl numberp = Pressureqi = Heat flux vector componentqi,SGS = Subgrid scale heat flux vector componentRe = Reynolds numberSij = Strain tensorSt = Strouhal numberT = Temperaturet = TimeUconv = Convection velocity of the vorticesui = Velocity componentxi = Space coordinatey+ = Wall viscous coordinate[5pt] Greek symbols∆ = Filter sizeγ = Ratio of specific heatsµt = Turbulent viscosityΦ = Flow variable

∗PhD Candidate, Department of Mechanical Engineering, Celestijnenlaan 300B, 3001 Leuven, Belgium. e-mail: gus-tavo.rubio@mech.kuleuven.be

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

ρ = Densityτij = Shear stress tensorτij,SGS = Subgrid scale shear stress tensorζ = Lag factor in the Rossiter frequencies formula

Operators(·) = Favre-filtered quantity(·) = Filtered quantity

Subscripts(·)∞ = Value at infinity

Superscripts(·)′ = Fluctuation value

I. Introduction

In exhaust ducts, expansion chambers are commonly installed to attenuate the noise emitted by e.g. IC en-gines and compressors. The energy of this engine noise is concentrated around the harmonics of the engine

firing frequency. Their contribution is dominant for low to medium engine speeds due to the relative ineffi-ciency of aerodynamically generated noise sources at low Mach number. When the engine speed increases,flow noise effects become more important and can even become the dominant source of exhaust noise1 . Inthis framework, expansion chambers can become flow-excited sound generators rather than silencers.

Expansion chambers are traditionally designed to damp out engine noise and flow noise sources, generatedat flow discontinuities or singularities inside the exhaust system upstream to the expansion chamber. Incontrast to internally generated flow noise, these sound attenuation principles are quite well understood2,3 .The purpose of the numerical simulation presented here, is to gain more insight in the internally generatedflow noise sources in expansion chambers. The tonal noise components can either be generated by a flow-acoustic feedback-loop inside the expansion chamber or by an excitation of chamber and tailpipe acousticmodes.

This paper discusses, for a simple 3D expansion chamber, the ability of compressible Large Eddy Sim-ulation to compute flow–acoustic interactions, and the numerical difficulties that arise for ComputationalAaeroacoustics applications in confined subsonic flows.

II. Noise sources in expansion chambers

A distinction between three different kinds of internal flow noise sources can be made. First, the turbu-lence inside the confined jet in the expansion chamber and tail pipe generates a broadband noise. Since thistype of flow noise source has a quadrupole nature with a radiated acoustic power that scales with M8, it is,for moderate Mach numbers, a less effective noise source than the other two sources, which are monopoleor dipole like (scaling with M4 resp. M6). The latter sources are generated by a flow-acoustic feedback-coupling inside the expansion chamber or by the excitation of the acoustic resonances of the chamber andthe tailpipe.

The basic mechanism of the flow-acoustic coupling is well-known4 : when a flow leaves a sharp backwardsfacing edge it separates and forms a thin shear layer or vortex sheet. Such sheets are very unstable andquickly roll up to form a train of vortices. In the case of the chamber resonances, the vortex sheet is perturbedby the acoustic waves reflected at the downstream edge of the expansion chamber. Based on experimentalresults on cavity flows, Rossiter5 derived the following semi-empirical formula for the Strouhal number St ofthis phenomenon:

St =fD

U∞=

n− ζ

M + 1/k(1)

with f the frequency, D the height of the inlet pipe, U∞ the free-stream velocity, n the mode number,M = U∞/c the undisturbed Mach number, c the speed of sound, k = Uconv/U∞ the ratio of the convectionvelocity of the vortices (Uconv) to the free-stream velocity and ζ a factor to account for the lag time betweenthe passage of a vortex and the emission of a sound pulse at the trailing edge of the cavity.

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Another possible source of tonal noise is the excitation of the purely acoustic resonances by the flow.The acoustic resonance frequencies for the tailpipe and the expansion chamber can be obtained using thefollowing formula1 :

St =fD

U∞= nD

(1−M2)2MLc

(2)

where Lc the characteristic acoustic length of the expansion chamber or tailpipe. For the expansion chamberLc = L, with L the length of the expansion chamber, for the tailpipe Lc = (Lt + l) where Lt is the lengthof the tailpipe and l ≈ 0.525D is an end-correction3 . This formula is only valid for 1D propagation ofacoustic waves. Above the first transversal resonance frequency of both tailpipe and expansion chamberother resonances occur due to the 3D acoustic propagation.

The remainder of the paper concentrates on purely internally generated flow noise. As such, the presentwork will not handle the noise sources generated upstream of the expansion chamber. Also broadband jetnoise generated at the exit of the tailpipe is not investigated, since this kind of noise is, at moderate Machnumbers, quite inefficient due to its quadrupole nature6 .

III. Problem description

In the present paper, a rectangular expansion chamber like the one shown in figure 1 is investigated.All the lengths are made non-dimensional with the height of the inlet pipe, D. The dimensions in the Zdirection are the same as in the Y direction. To avoid possible coincidences in resonance frequencies, thelength of the inlet and outlet pipes are different from the length of the chamber. In this way, it will be easierto differentiate in the results the origin of the acoustic excitations.

Figure 1. Computational domain

The Reynolds number of the flow is 1000, based on the height D of the inlet pipe and on the maximumvelocity at the inlet. The Mach number is equal to 0.5. Although this Mach number is unrealisticallyhigh for this type of applications, it has been chosen in order to increase the time step and to reduce thecomputational time. More realistic configurations (higher Reynolds number, lower Mach) will be investigatedin the future and compared with experiments.

A structured multi-block mesh has been constructed consisting of approximately 2.986.464 cells. Theconnectivity between the blocks has been done using full matching boundary conditions.

IV. Numerical modeling

The compressible LES equations for a viscous flow are obtained from a decomposition of the variablesof the Navier-Stokes equations (ρ, ui) into a filtered part (ρ, ui) and an unresolved part (ρ′, u′i) that hasto be modeled with a subgrid scale model. In most LES of compressible flows, the flow variables are Favreaveraged or density weighted

Φ =ρΦρ

(3)

where the bar denotes the LES filtering, ρ is the density and Φ is any given flow variable. Using thisapproach, the following equations are obtained for, respectively, mass, momentum and energy:

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∂ρ

∂t+

∂ρui

∂xi= 0 (4)

∂ρui

∂t+

∂ρuiuj

∂xj= − ∂p

∂xi+

∂(τij + τij,SGS)∂xj

(5)

∂ρe

∂t+

∂ρeui

∂xi= −∂uip

∂xi+

∂ui(τij + τij,SGS)∂xj

− ∂(qi + qi,SGS)∂xi

(6)

where ρ, ui and p are the resolved density, velocity components and pressure. For a perfect gas, the totalenergy per mass unit e is defined as p/(γ − 1)ρ + (u2

1 + u22)/2 with γ the ratio of specific heats.

The viscous stress tensor τij is modeled as a Newtonian fluid, and the heat flux qi is modeled withFourier’s law. Dynamic molecular viscosity and molecular conductivity are kept constant. The subgrid scalestress tensor τij,SGS , and the subgrid scale heat flux qi,SGS , reproduce the dissipative effects of the unresolvedscales by using a turbulent viscosity µt, and a turbulent Prandtl number Prt. These are then modeled as

τij,SGS = 2µtSij (7)

and

qi,SGS = −(

µtcp

Prt

)∂T

∂xi(8)

with T the temperature and Sij the resolved rate of strain tensor

Sij =(

12

) (∂ui

∂xj+

∂uj

∂xi

)(9)

To determine the turbulent viscosity, a Smagorinsky model is used, where

µt = ρ(Cs∆)2√

2SijSij (10)

The Smagorinsky constant Cs and the turbulent Prandtl number are set to 0.12 and 0.5, respectively.The filter size ∆ is locally set to the cube root of the cell volume. To take into account the scale reductionthat occurs near walls, the filter size ∆ is weighted with the normal wall coordinate y+ in the way proposedby Van Driest7 : ∆′ = ∆(1− exp(−y+/A)).

The main reason to proceed this way is that the Favre averaged equations are simpler, since in this case,no model is needed in the continuity equation. Although by using Favre averaging of the momentum equationsome information can be lost indeed, Freund et al.8 have shown that, for relatively low Mach number, thedifference between the Favre and the non-Favre averaged equations are small for mixing layer flows.

However, the Favre averaged continuity equation 4 is still a non–linear equation which can cause numericalinstabilities. In the test–case under study, such instabilities are observed in the inlet and outlet pipes, leadingto grid–to–grid oscillations. These oscillations can be removed in several ways. Visbal et al.9 , apply animplicit spatial filter at the end of every time integration step. The filter is applied to the conservativevariables sequentially one direction at a time, in order to minimize possible bias of the solution. Anotherway to remove grid–to–grid oscillations is to use the non–Favre averaged LES equations. In the derivationof this set of equations, a damping term appears in the right hand side of the continuity equation, whichhelps to stabilize the solution. However, the appearance of an unsteady subgrid term in the momentumequation10 makes the implementation of these set of equations more complex. The easiest way to stabilizethe solution and to avoid the appearance of grid–to–grid oscillations is to add an artificial diffusion term inthe right hand side of the continuity equation and this approach is chosen in this paper. Hereby dissipationis based on the extra term appearing in the right hand side of the equation when a non–Favre average isused during the filtering operation. This term has the form10 :

− ∂(ρui − ρ ui)∂xi

(11)

With the help of the Boussinesq hypothesis, a simple model can be written for this subgrid mass fluxthat can be implemented as artificial dissipation for the continuity equation.

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(ρui − ρ ui) = −Cρ∆2

√2SijSij

∂ρ

∂xi(12)

The value of the constant Cρ is set to 0.1The filtered compressible Navier–Stokes equations are implemented in a finite volume code. They are

integrated in time using a fourth-order explicit Runge-Kutta scheme. Convective and viscous terms arediscretized using central second-order schemes. At the inlet, velocity profiles and static temperature havebeen imposed. At the outlet, static pressure has been prescribed. No penetration and no slip boundaryconditions have been applied in all isothermal walls. The code has been fully parallelized with dynamic loadbalancing using MPI.

V. Results

A. Flow results

The computation was performed for 21 non–dimensional time units (1.077e-5 seconds of real time) beforestarting averaging and recording time data. This represents 1.6 through–flow times. The flow has beenaveraged for another 20 non–dimensional time units (1.027e-5 seconds or 1.54 through–flow times). Afterthis, the averaged non–dimensional mass flow in the inlet is 1.290121 and in the outlet 1.272392, whichmeans that the mass flow rate is converged to an error of 1.3%.

In figure 2, a vorticity iso–surface for the statistically converged flown is shown. Even if the Reynoldsnumber is quite low, the shear layers start to roll up very soon and break down into a periodic vortexshedding. These vortices are convected downstream in the expansion chamber until they hit the oppositewall of the chamber where they break down into smaller vortices. Some of this turbulent kinetic energy istrapped in semi-stationary vortices in the corners of the expansion chamber, and the rest is convected awaydown the outlet pipe and leaves the domain.

Figure 2. Vorticity iso–surface colored by U-velocity.

The vortex shedding from the inlet pipe can be noticed also from the instantaneous pressure iso–contoursin the center plane shown in figure 3, where the low pressure regions in the center of the vortices becomeevident.

In figure 4, time–averaged turbulent kinetic energy is shown in several cuts perpendicular to the directionof the flow. The most important sources of turbulent kinetic energy are the shear layers created from the

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Figure 3. Instantaneous pressure contours. 50 contours of p/ρU2∞ are shown between 3 and 4.

interaction of the flow coming from the inlet pipe with the stagnant flow inside the chamber. The turbulentkinetic energy reaches a maximum near the entrance of the tail pipe, which is the region where aeroacousticsources are expected to be dominant. Once the flow enters the tail pipe, turbulent kinetic energy starts to bedissipated at a high rate. The length of the tail pipe is important in two ways. Firstly, the velocity magnitudeat the tail pipe exit will be dependent on the pipe length, and secondly, the strength of aeroacoustic sourcesgenerated at the interaction between the discharging jet and the atmosphere will also depend on the amountof turbulent kinetic energy, dissipated between the expansion chamber and the outlet.

Figure 4 shows also the time–averaged statistical U–velocity and pressure extracted from the domaincenterline. In both pipes, the flow behaves the same, the pressure decays linearly and the flow acceleratesas expected in these channel flow type. When the flow enters the expansion chamber it expands, keeping aconstant pressure and velocity over the whole length of the expansion chamber, until it reaches the contractionat the entrance of the outlet pipe, where there is a sudden drop in static pressure, making the flow to undergoa sudden acceleration.

Figure 4. Time–averaged turbulent kinetic energy (left) and time–averaged pressure and U–velocity along thecenterline of the domain (right).

Two main sources of aeroacoustic noise can be identified from the flow results. The first one is the roll upof vorticity in the shear layer generated when the flow enters the expansion chamber from the inlet pipe. Thissource is probably more effective for tonal noise, since the vortex shedding happens at a specific frequencygiven by the flow–acoustic interaction. The second one is the rear wall of the cavity in the region around theentrance to the tail pipe. Here, vortices impact on the wall breaking up and generating a strong exchange ofenergy between the flow and the acoustics. This source is expected to be more effective for broadband noise.

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B. Acoustic results

To study the acoustic behavior of the chamber, the time history of density, velocity and pressure are recordedin 4 different points inside the domain, 3 inside the chamber and 1 inside the tail pipe. The exact locationof these points is given in table 1.

Table 1. Location of the monitoring points.

Point X Y Z Location1 0.12 -1.15 -1.15 Chamber2 2.86 0.14 0.14 Chamber3 4.75 1.32 1.32 Chamber4 7.32 0.1 0.1 Tail pipe

As explained in section II, three types of resonance can occur in the domain: Rossiter frequencies,chamber resonances and tail pipe resonances. The first acoustic resonance for this chamber at Mach 0.5 isexpected, according to equation 2, at a Strouhal number of 0.15. Also, according to equation 1, the firstRossiter frequency should occur at a Strouhal number of 0.33. Figure 5 shows the pressure response spectrain the 3 points inside the chamber.

Figure 5. Pressure response spectra inside the chamber.

The 3 spectra exhibit a chamber resonance at a Strouhal number of 0.215 with points 1 and 3 (see table 1)showing a higher amplitude than point 2. The latter is explained by the fact that the first chamber resonancehas its maximum near the front and back walls of the chamber and its minimum at the center.

To understand the discrepancy between the expected value and the computed value, it is important tonote that the formula for these resonances is dependent on the Mach number which shows a great variabilityinside the domain. Figure 6 shows the average Mach number in the center plane of the domain.

The explanation for the difference is that the Mach number of 0.5 is imposed at the inlet. A Strouhalnumber of 0.215 corresponds to a Mach number of 0.39. When the flow enters the expansion chamber itexpands and decelerates, giving an average Mach number inside the chamber lower than 0.5.

The response spectrum at point 2 exhibits, apart from the chamber resonances, another peak at a Strouhalnumber of 0.3. This is the first Rossiter mode. This frequency is associated with the periodic vortex sheddingand its interaction with acoustic waves. Since this point is the only one located in the region of influence ofthe shear layers, this is the only point that show this type of resonance.

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Figure 6. Average Mach number in the center plane.

Figure 7 shows the response spectrum found in the point located inside the tail pipe.

Figure 7. Pressure response spectrum inside the tail pipe.

According to equation 2 a resonance is expected at Strouhal number 0.166 for a Mach number of 0.5.The computed value is 0.086. This value corresponds to a Mach number of 0.685. Looking at figure 6, it isclear that the average Mach number inside the tail pipe is around this value. The second peak in figure 7at Strouhal 0.23 is higher than the first one. This can be explained because of a coincidence between thesecond tail pipe mode and the first chamber mode. However, the resolution of the spectrum does not allowto see this clearly. In order to have a better resolution, the time signals should be recorded for a longer time.

VI. Conclusions and future work

Aeroacoustic noise source mechanisms have been investigated for a simple expansion chamber. Theacoustic resonances are predicted in the chamber and in the outlet pipe. It is clear that, when usingempirical/analytical expressions to estimate these frequencies, special care has to be taken in order toestimate correctly the local Mach number. Inside the expansion chamber, also the first Rossiter mode isaccurately predicted. The two main mechanisms of sound generation identified are the roll up of vorticityand break up of the shear layer as the flow enters the chamber (tonal noise), and the interaction of the

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convected vortices with the rear chamber wall, where a strong exchange of flow-acoustic energy happens.Acoustic resonances of both, the expansion chamber and tail pipe, are excited by the flow field. It is clearthat LES has the ability to predict tonal flow noise frequencies. The work presented in this paper is thefirst step in the development of a numerical methodology to study aeroacoustics in subsonic confined flows.Further research will focus on experimental validation of the results, and on the computation of more realisticconfigurations at higher Reynolds and lower Mach. The numerical methodology will be also improved byincreasing the order of the numerical schemes, investigating the interaction between the resolved scales anddifferent subgrid scale models, and improving boundary treatments.

Acknowledgements

The research work of Wim De Roeck is financed by a scholarship of the Institute for the Promotion ofInnovation by Science and Technology in Flanders (IWT).

References

1J. M. Desantes, A. J. T. and Broath, A., “Experiments on Flow Noise Generation in Simple Exhaust Geometries,” ActaAcustica, 2001.

2Munjal, M. L., Acoustics of Ducts and Mufflers, Wiley–Interscience, New York, 1987.3Davies, P. A. O. L., “Practical Flwo Duct Acoustics,” Journal of Sound and Vibration, 1988.4Davies, P. A. O. L., “Flow–Acoustic Coupling in Ducts,” Journal of Sound and Vibration, 1981.5Rossiter, J. E., “Wind Tunnel Experiments on the Flow over Rectangular Cavities at Subsonic and Transonic Speeds,”

Technical report 64037, Royal Aircraft Establishment, 1964.6A. J. Torregrosa, A. Broath, H. C. and Anders, I., “A Note on the Strouhal Number Dependance of the Relative

Importance on Internal and External Flow Noise Sources in IC Engine Exhaust Systems,” Journal of Sound and Vibration,2005.

7Driest, E. R. V., “On Turbulent Flow near a Wall,” Journal of Aeronautical Science, 1956.8J. B. Freund, P. M. and Lele, S. K., “Compressibility Effects in a Turbulent Annular Mixing Layer.” Technical report

tf-72, Stanford University, 1997.9Visbal, M. R. and Gaitonde, D. V., “High–Order–Accurate Methods for Complex Unsteady Subsonic Flows,” AIAA

Journal , 1999.10Boersma, B. J. and Lele, S. K., “Large Eddy Simulation of Compressible Turbulent Jets.” Annual research briefs, Center

for Turbulence Research, 1999.

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