CALCULATIONOF THE TAIL-PIPE NOISE OF EXHAUST SYSTEMS

WITH WAVE

Rolf Jebasinski

J. Eberspächer

73730 Esslingen

Germany

ABSTRACTIn the past the calculation of exhaust systems tail-

pipe noise was a problem which was onlyunsatisfactorily solved. The following paper shows thepossibilities of engine simulation programs based onone-dimensional Computational Fluid Dynamicscodes, such as WAVE, to calculate the tail-pipe noiseof exhaust systems. Comparison of simulation andmeasurement show that it is possible to accuratelypredict tail-pipe noise if the muffler, with all interiorpiping systems, is correctly modeled. Even 3-dimensional effects can be simulated with a specialmodeling approach. In addition the backpressure canbe calculated so that the whole exhaust system can beoptimized with respect to tail-pipe noise andbackpressure.With the new preprocessor KADOS for WAVE,which significantly reduces the time for modelingexhaust systems with WAVE, new potentials arise forthe development and optimization of exhaust systems.

INTRODUCTIONThe time-to-market time for new automobiles has

decreased in the last years considerably. For this rea-

son the suppliers have to develop their products inshorter time too. One possibility to approach this goalare increased simulations of the specific character-istics of the product to reduce cost-intensive proto-typing and tests.Besides emission controls the noise attenuation is themost important function of engine exhaust systems.With increasingly stringent legislation and regulationsthe demands made on exhaust systems increaseconstantly.The performance of an exhaust system is measured interms of tail-pipe noise, which is the sound pressurelevel in a short distance to the tail-pipe. Differentconcepts have been tested at Eberspächer over thepast, which promised a good prediction of the tail-pipenoise.Each development process is complicated by conflict-ing targets. In the case of exhaust systems these arethe demands for optimum noise reduction whilekeeping backpressure at a minimum. In additioncertain package constraint exists, i.e. the position andsize of the mufflers are mostly fixed. The acousticoptimization can therefore only be made withvariation of the duct diameter and the interior piping

system of the mufflers. Calculation methods shouldtherefore be able to predict the tail-pipe noise and thebackpressure for a complete exhaust system, includingmufflers with complex interior piping system(perforated ducts and baffles).Possible calculation methods are the Transfer MatrixMethod, Finite Elements Method (FEM), BoundaryElements Method (BEM) or Computational FluidDynamics Method ( CFD ).The most common calculation method is the TransferMatrix Method (or four-pole theory). This method isbased on the linear one-dimensional wave propagationin ducts and formulation of the individual elementslike ducts, area discontinuities and branches intransfer matrices in analogy to the electric filter theory[1,2].Noise attenuation of simple mufflers can be calculatedwith this method in frequency domain. The assump-tion of linear wave propagation restricts this method toparts of the exhaust system where the sound pressurelevel is less than 160 dB (i.e. down pipe, exhaustmanifold and catalyst cannot be calculated).For the prediction of the tail-pipe noise the sourceimpedance (which is given by engine and exhaustmanifold) must be known, either through measure-ments or calculations.This method has several disadvantages. It is not pos-sible to calculate the backpressure with it. The engineimpedance is usually measured independent of thegiven exhaust system. The response of the exhaustsystem on the engine thus remains unconsidered.Besides that the calculation of perforated ducts is notsolved completely. There are several competingmodels for the description by perforated ducts[3,4,5,6], which lead to different results (for acomparison see [7]). Additionally these models are allbased on experimentally determined impedance’s ofthe perforation, since non-linear effects play a role.This non-linear behavior of the impedance appears

after Sullivan [8] already at a sound pressure level of130 dB. Such levels are reached in nearly every rearmuffler.One-dimensional CFD-programs, which also simulatethe thermodynamic processes in the engine, likePROMO [9] and WAVE [10], can help here.With these programs the insteady, non-linear gas flowcan be calculated in the entire exhaust systemincluding engine. One receives thus without detoursthe tail-pipe noise under consideration of non-lineareffects in the exhaust system, as well as the backpres-sure of individual components in the exhaust system.This contribution shows some comparisons betweenmeasurements and WAVE-calculations of exhaustsystems tail-pipe noise.

ONE-DIMENSIONAL MODELING OFMUFFLERS WITH WAVE

For the simulation of an exhaust system acorresponding WAVE - model must be prepared. Forthis purpose WAVE offers two main elements, ductsand junctions. Junctions are volumes, which can beconnected to each other and to ducts. The followingsection describes how mufflers are modeled with theseelements.First we will take a very simple example, a concentrictube resonator. This muffler has a perforated duct thatis enclosed by an outer can. The muffler is subdividedinto segments of length ∆x. Each segment correspondsto a volume or junction for the duct and the outerresonator. Fig. 1 (b) shows this schematically.The junctions representing the duct and the resonatorare connected in flow direction correspondingly. Inaddition a connection exists between the junctions ofthe duct and the resonator via a perforated wall.The segment length ∆x determines the frequencyresolution as well as the computing time. The smallerthe segments are, the higher the frequency resolutionbut unfortunately the computing time too.

∆x

a)

b)

FIGURE 1:Concentric tube resonator (a) and WAVEmodel (b), grey junctions represent the perforatedtube and white junctions the resonator.

With a segment length of 40 mm one reaches a goodprecision up to 600 Hz. The computing time lies inthis case for a complete model of an exhaust system(inclusively simple intake system and engine) at 10speed points on a RISC-Workstation by approximately1-2 hours.Generally the interior piping system of a muffler ismuch more complex than in the above-mentionedexample. Then the muffler must be subdivided in moresegments. Fig. 2 presents an example of a typicalseries rear muffler.The modeling of a complete exhaust system withmanual entry in the WAVE input file can take severaldays depending on the complexity of the muffler.Changes in the geometry of the muffler, i.e. throughdisplacement of a baffle or change of a duct diameter,lead to a recalculation of the cross-sections andvolumes of the junctions and ducts and a new inputinto the WAVE file.Optimization of mufflers would therefore be very timeconsuming.

FIGURE 2: rear muffler 1 (a) and WAVE model (b)

To simplify this entry and to shorten the developmentprocess the graphic preprocessor KADOS(Knowledge Based Automated Design of Intake andExhaust Systems) for WAVE was developed by aninternational consortium under participation ofEberspächer.KADOS offers different elements (perforated ducts,bent ducts, baffles etc.) which can be placed in themuffler with the computer mouse as in drawingprograms. Fig. 3 (b) shows the geometry input maskfor oval-end mufflers. Fig. 3 (a) shows the CAD -model of the rear muffler sketched in Fig. 3 (b).After establishing the model of the exhaust system onthe screen KADOS generates a WAVE input file.Changes in geometry or in the interior piping systemof mufflers can now be made very quickly.

FIGURE 3: rear muffler 2 (a) and KADOS inputmask (b).

ResultsMeasurement and calculation of the tail-pipe noise

were done on a Mercedes Benz C200 (4 cylinder 2liter SI engine) with the series exhaust system. Theexhaust system consists of a catalyst with twomonoliths, middle muffler (simple concentricresonator with two chambers) and rear muffler 1 (seeFig. 2). The engine data (Combustion profile, valvetiming etc.) were taken from Ref. [11].Tail-pipe noise was measured and calculated at adistance of 22 cm. The experimental conditions werefull load acceleration. Since WAVE is not able tosimulate absorption material, the absorption materialin both mufflers was removed.

Fig. 4 shows the tail-pipe noise of measurement andWAVE simulation. Up To 3000 rev/min the curvesagree very well. The calculated level lies only 3 dB(A)below the measured level.

1 00 0 20 0 0 3 0 0 0 4 0 00 5 00 0 60 0 08 0

8 5

9 0

9 5

10 0

10 5

11 0

11 5

measurement WAVE calculation WAVE calculation + flow generated noise WAVE calculation (mufflers modeled as expansion chambers)

sou

nd

pre

ssu

re le

vel [

dB

(A)]

s p e ed [re v /m in ]

FIGURE 4: Tail-pipe noise rear muffler 1,measurement and WAVE calculation, A-weighted.

A calculation with simple expansion chambers insteadof the original mufflers is shown in Fig. 4 for com-parison. The tail-pipe noise of this configurationdeviates clearly from the experimental results and thecalculation with the original muffler.At 3000 rev/min flow generated noise becomesdominant, which lead to a difference betweenmeasured and calculated values.Besides sound pressure the flow velocity and the gastemperature are known from the WAVE calculation.Thus the flow generated noise can be calculated withthe empirical formula from Green and Smith [12].According to this formula the sound power level (Lw)depends on the flow velocity (v), gas temperature (T),diameter of the inlet tube (D) and the efficiency factorof the rear muffler (Ew).

Lw = Ew - 17.5*logT + 20*log(D) + 45*log(v) +1,87

Efficiency factors were taken from Ref. [12]. Thecurve thus calculated is displayed in Fig. 4. Theagreement is good for such simple assumptions.At 5500 rev/min the calculated backpressure of115 mbar lies somewhat over the measured value of95 mbar (see also Table 1).

1000 2000 3000 4000 5000 600070

80

90

100

110

speed [rev /m in ]

70

80

90

100

110

soun

d pr

essu

re le

vel [

dB(A

)] 70

80

90

100

110

6. engine order

4. engine order

2. engine order

measurement WAVE calculation

FIGURE 5: Sound pressure level of the 2.,4. and 6.engine order of rear muffler 1, A-weighted.

Fig. 5 shows the level of 2. , 4. and 6. engine orders,which dominate the tail-pipe noise. The measured andcalculated values of the 2. engine order are in excel-lent agreement. At 4. and 6. engine order the agree-ment is good in the lower speed range. The calculated6. engine order shows a small shift in speed comparedto the measurements. The reason for this is still objectof further investigations. At higher speed andtherefore higher frequencies flow generated noise

cause larger deviation between measurement andcalculation.Fig. 6 shows WAVE-calculated third-octave spectra at2500 rev/min and measured values. The agreement isgood.

80

90

1002500 rev/min

ress

ure

leve

l [dB

(A)]

FIGURE 6: Third-octave spectra of rear muffler 1 at2500 rev/min, A-weighted.

For further comparisons rear muffler 1 was replacedby rear muffler 2 (shown in Fig. 3 (a)).In Fig. 7 the level difference between both mufflers inthe dominating engine order are presented. Theagreement between measurement and calculation isgood.Above 3000 rev/min the level differences at the 4. and6. order show some deviation between measurementand calculation, which could be due to flow generatednoise.For the second rear muffler the WAVE calculationyielded a backpressure of 102 mbar in comparisonwith measured 87 mbar.A comparison of the backpressures show that therelative change of the backpressure can be determinedwith the WAVE calculation. The absolute values lieapproximately 15 % above the measured values.

1000 2000 3000 4000 5000-28-24-20-16-12

-8-40

speed [rev/m in ]

-20

-16

-12

-8

-4

0

leve

l diff

ere

nce

[dB

]

-20

-16

-12

-8

-4

0

6. engine order

4. engine order

measurement WAVE calculation

2. engine order

FIGURE 7: Level difference between rear muffler 1and rear muffler 2; 2., 4. and 6. engine order.

Table 1:WAVE - calculation measurement

rear muffler 1 115 mbar 95 mbarrear muffler 2 102 mbar 87 mbar

WAVE is thus very well suited for the optimization ofmufflers, with regard to noise attenuation and back-pressure, through modifications of the interior pipingsystem.A further example will illustrate this. In a currentproject a muffler for a V6 diesel truck is beingdeveloped. The tail-pipe noise of this vehicle wasdominated through the 3. engine order.Especially at the rated speed of 1800 rev/min veryhigh sound pressure levels were observed with thefirst design of the muffler. With the calculation the

attempt was made to modify the muffler withoutchange in volume and to reduce the 3. engine order.Fig.8 shows the level difference between initial designand modified design, both calculated by WAVE.Through an additional perforation in the inlet pipe areduction of 4 dB at 1800 rev/min has been reached inagreement with the measurements.

900 1000 1100 1200 1300 1400 1500 1600 1700 1800-5

-4

-3

-2

-1

0 m easurem ent

WA V E calculation

leve

l diff

eren

ce 3

. eng

ine

orde

r [dB

]

speed [rev/min]

FIGURE 8: Level difference between optimizedmuffler and original muffler for a truck; 3. engineorder.

3-dimensional modeling of mufflersThe investigations described in the previous para-

graphs were based on one-dimensional modeledmufflers. So the frequency resolution is limited toapproximately 600 - 1000 Hz depending on thesegmentation length ∆x in flow direction.

It will be shown in this paragraph that is possible toextend this frequency limit much farther by using a 3-dimensional mesh for the muffler under investigation.The simulations will be compared to TransmissionLoss measurements from Selamet et al. [13].Fig. 9 (b) shows the 1-dimensional mesh of a simpleexpansion chamber. Each segment is a volume. Thedimensions of the chamber are shown in Fig. 9 (a).

FIGURE 9: Dimensions of the expansion chamber(a) 1-dimensional WAVE-model (b) and3-dimensional WAVE-model (c).

The first radial cross mode propagates at a frequencyof 2737 Hz. Nevertheless multidimensional behaviorcan be observed at lower frequencies depending on thelength/diameter ratio as shown in Ref. [13]. Fig. 10displays the calculated Transmission Loss for a 1-dimensional WAVE model of this muffler. Forcomparison the measured Transmission Loss and theresult of a 3-dimensional BEM calculation, adaptedfrom Ref.[13], is printed in Fig. 10.The 1-dimensional WAVE calculation deviates clearlyfrom the measurements and the BEM calculation atfrequencies above 2000 Hz. Minor differences can beobserved between 1000 and 2000 Hz.

0 500 1000 1500 2000 2500 30000

10

20

30

40

50

m easurem ent

B E M calculation

WA V E with ∆ x = 31 m m

Tran

smis

sion

Los

s [d

B]

Frequency [Hz]

FIGURE 10: Transmission Loss measurement, 3-dimensional BEM calculation and 1-dimensionalWAVE-calculation of the expansion chamber fromFig. 9.

Therefore the modeling of the expansion chamberwith WAVE was revised. In addition to thesegmentation in x-direction a second segmentation inradial direction was made. This can be accomplishedby the “YJunction“ element in WAVE. Because of therotational symmetry a radial segmentation is sufficientfor a 3-dimensional WAVE mesh. Fig. 11 shows theTransmission Loss of this model (∆x = 31mm and ∆r= 30 mm).The deviation between calculation and measurement ismuch smaller. The peak at 2500 Hz can be observednow, however with a shift to lower frequencies.To improve this result the segmentation length in x-and r-direction has been varied (see Fig. 11). Reducingonly one segmentation length gives worse results,whereas decreasing ∆x and ∆r by an equal amountyields a better result as can be seen in Fig. 11. In thecase of ∆x=15 mm and ∆r=15 mm the predicted peakis very close to the measured one and the minimum at2200 Hz is captured well. So it seemed to be importantto use equal segmentation length in every direction tocapture 3-dimensional effects.

0 500 1000 1500 2000 2500 30000

10

20

30

40

50 m easurem ent

WA V E ∆ x = 31 mm ,∆ r = 30 m m

WA V E ∆ x = 15 mm ,∆ r = 30 m m

WA V E ∆ x = 31 mm ,∆ r = 15 m m

WA V E ∆ x = 15 mm , ∆ r = 15 mm

Frequency [dB]

Tran

smis

sion

Los

s [d

B]

FIGURE 11: Transmission Loss calculations withdifferent segmentation length ∆∆∆∆r and ∆∆∆∆x in theWAVE-model.

To support this a second expansion chamber wasmodeled, which had a length of 94 mm. The result ofthe WAVE model with segmentation lengths of 15 and30 mm in both directions is plotted in Fig. 12 togetherwith the measured results. This example shows thatwell beneath the cut-off frequency of the first radialmode multidimensional wave propagation is apparent.

0 500 1000 1500 2000 2500 30000

5

10

15

20

25 m easurem ent

WA V E 1-D ∆ x = 31 m m

WA V E 3-D ∆ x= 31 mm , ∆ r= 30 m m

WA V E 3-D ∆ x= 15 mm , ∆ r= 15 m m

Tran

smis

sion

Los

s [d

B]

Frequency [Hz]

Figure 12: Transmission Loss calculation withWAVE of an expansion chamber with l=93 mm.

Up to now these 3-dimensional effects were onlyaccessible with 3-dimensional simulation methods asBEM and FEM. With the “YJunction“ WAVE offersan element that can be used to make a quasi 3-dimen-sional mesh of a muffler, so that the 3-dimensionalbehavior of the sound pressure waves can becalculated with a 1-dimensional CFD Code.This method offers the advantage of enabling non-linear calculation of the insteady flow excited by anengine. This cannot be done with BEM and FEM,since they are purely acoustic calculation methods(i.e. linear without inclusion of flow).The longer computation times, compared to the1-dimensional modeling approach could be adisadvantage.

SummaryThe comparison of measurement and simulation on

an exhaust system in a series vehicle has shown thatone can receive a good prediction of the dominatingengine orders with WAVE by using a 1-dimensionalmodeling approach.Thus level differences between different rear mufflersare well predicted in the lower speed range (up to3000 rev/min).In the higher speed range (above 3000 rev/min) strongflow generated noise sets in. This flow generatednoise can be determined too with a simple empiricalformula, and the gas velocity and gas temperatureobtained from the WAVE calculation.Since beside the sound pressure level the backpressurecan be deduced from the WAVE results, WAVE iswell suited as supporting tool for the development andoptimization of exhaust systems.With the new graphic preprocessors KADOS themodeling of exhaust system has been simplified,which reduces the time for the establishing of aWAVE model considerably.

New possibilities are offered by adapting the 3-dimensional modeling approach, presented in the lastparagraph, into KADOS, which might extend thefrequency resolution into the kHz range.

References

[1] M.L. Munjal, Acoustics of Ducts and Mufflers, John Wiley & Sons, Inc., 1987

[2] K. Lehringer, Automobil Industrie, 1988, 6[3] J.W. Sullivan and M.J. Crocker, Journal of the

Acoustical Society of America 64, 1978, 207[4] J.W. Sullivan, , Journal of the Acoustical Society

of America 66, 1979, 772[5] K. Jayaraman and K. Yam, Journal of the

Acoustical Society of America 69, 1981, 390[6] K.N. Rao and M.L. Munjal, Proceedings of the

1984 Nelson Acoustics Conference,Madison WC

[7] K.S. Peat, Journal of Sound and Vibration 123, 1988, 199

[8] J.W. Sullivan, , Journal of the Acoustical Society of America 66, 1979, 779

[9] H. Seifert, Motortechnische Zeitschrift 51,1990, 11

[10] T. Morel, M.F. Fleming and L. LaPointe,SAE 900679

[11] J. Abthoff, D. Hüttebräucker, W. Zahn and H. Bockel, Motortechnische Zeitschrift 53,1992, 496

[12] A.J. Green and P.N. Smith, IMechE 1988, C17/88, 47

[13] A. Selamet and P.M. Radavich,SAE 950544