Sonderforschungsbereich/Transregio 40 – Annual Report 2009 165

Influence of Scaling Rules on the Loss ofAcoustic Energy through Demonstrator

Rocket Nozzles

By D. Morgenweck, F. Fassl † AND T. SattelmayerLehrstuhl für Thermodynamik, Technische Universität München

Boltzmannstr. 15, 85748 Garching b. München

The damping of acoustic oscillations in two different rocket engine demonstrators isassessed. Both kinds of engines represent different scaling strategies for sub-scaledemonstrators. In this context one major question is, which of the two designs behaveless prone to combustion instabilities. In the following different methods are presentedto show the influence of geometry on the loss of acoustic energy. The decay coefficientis determined, which accounts for the overall loss of acoustic energy independent of thephysical mechanism. The admittance at the intersection of combustion chamber andnozzle is evaluated. The acoustic flux is calculated and divided into contributions of pureacoustic transport and convective transport. The importance of convective transport isemphasized by analyzing the relation between admittance and acoustic flux. Anotherpart of the results of the stability analysis shows the significance of the character of themode.

1. IntroductionToday oscillating instabilities are still a major threat for liquid rocket engines. High fre-

quency oscillations can cause structural damage to the nozzle and combustion chamberas well as to the periphery. The increased heat exchange to the wall can cause thermalfailure and thus threatens the success of the mission. Since the beginning of modernrocket motor design there is the need to understand the mechanism of those instabili-ties. Crocco [1] did a fundamental investigation in this field at a very early stage. It wasfound that the main criteria for stability of the combustion process in a rocket engine isthe ratio between the input of acoustic energy by the combustion process and the lossof acoustic energy basically by transport through the nozzle or damping devices. Com-pared to that the loss of acoustic energy in the boundary layer close to the wall has onlya small contribution. Marble and Candel [2] derived an analytic relation for admittancesof short nozzles depending only on the Mach number. Thereby, admittance character-izes the reflection behavior of the nozzle and thus the acoustic energy flux through thenozzle to a certain degree. In a more sophisticated semi-numerical and in an experi-mental approach Bell and Zinn [3–5] investigated nozzle admittances for 3D acousticwaves and for nozzles of finite length. Although the admittance itself depends on theMach number it does not describe the acoustic flux in the presence of steady mean flowsufficiently. Cantrell and Hart [6] gave a formulation for acoustic energy in an isentropic

† Astrium GmbH, Business Division Launcher Propulsion, 81663 München, Germany

166 D. Morgenweck, F. Fassl & T. Sattelmayer

potential flow. Similar relations were also found by Morfey [7] and Zinn [4]. They showedthat there is an additional contribution to the acoustic energy transport related to theMach number.

The following work uses several characteristic quantities to describe the acoustic en-ergy loss of the rocket engine, namely admittance, decay coefficient and acoustic flux.The procedure is similar to the one used by Pieringer [8]. Two different demonstratornozzle designs are compared by means of these characteristic quantities. Both designswere derived from the actual rocket engine by different scaling rules. It is assumed thatboth designs show also a different acoustic behavior. To support the decision, whichdemonstrator is less prone to combustion instabilities, an acoustic analysis is carriedout. The two different designs are compared with respect to their acoustic characteris-tics.

2. Numerical methodFor the numerical evaluation given in this report the time dependent solver PIANO-

SAT is used. The base code PIANO is provided by the DLR Braunschweig [9]. Specialadaptions for stability assessment in rocket nozzles are applied, such as a term for cou-pling heat release fluctuation and acoustic perturbation. Moreover, special boundaryconditions are developed to allow the modeling of the injector plane and the absorbersin the combustion chamber. PIANO-SAT works in the time domain and utilizes spe-cial discretization schemes commonly used in the field of aero-acoustics. It is a timeexplicit finite difference scheme of higher order. To assess the onset of acoustic oscilla-tions in operating rocket engines a simple model for coupling combustion fluctuation andacoustic fluctuation was implemented [10]. The numerical method solves the LinearizedEuler Equations (LEE) or the Acoustic Perturbation Equations (APE) [11]. Thereby, theprimitive quantities for density, pressure and velocity are split into a mean part and aperturbation part. PIANO-SAT only solves the perturbation part. The mean quantitiesserve as input for the solver and originate from a CFD computation.

The investigations below are based on the LEEs.

3. Thrust chamber geometry and mean quantitiesAstrium GmbH investigates a new thrust chamber design. For evaluating and inves-

tigating the new design a sub-scale rocket engine is developed, that could be operatedas test rig. Two different scaling approaches are discussed regarding the transferabilityof the demonstrator test campaign results to the master design.

(a) geometrical scaling or photo scaling– Radial and axial length are scaled with the same scaling factor. The sub-scalethrust chamber is shorter and thinner than the master design.– The scaled nozzle is extended by a cylindrical part to assure a similar degree ofburning (l∗).– It constitutes the classical approach for constructing a sub-scale demonstrator.– The photo scaled design is depicted in figure 1(a).

(b) Mach number similarity or Mach scaling– Radial dimension is scaled in order to have a similar Mach number at the axis ofsymmetry as in the master design.– Flow field and boundary layer are similar to the master design.

Influence of Scaling Rules in Demonstrator Rocket Nozzles 167

(a) Photo scaled (b) Mach scaled

FIGURE 1. Thrust chamber with planes for the computation of the acoustic fluxes

– Thus, also the heat load is comparable to the master design.– The Mach scaled design is depicted in figure 1(b).

Figures 1(a) and 1(b) show also the planes used for the evaluation of the acoustic inten-sity in the study. In both thrust chambers these planes reside at the same axial distancefrom the injector plane (baseplate). In the numerical acoustic analysis pressure and ve-locity fluctuations are recorded over time at the circumference of the planes. The firstplane is close to the injector head plate. A second plane resides where the transitionbetween cylindrical part and nozzle may be assumed in the Mach scaled design (atx=0.1 m). The third plane marks the interface between nozzle and cylindrical combus-tion chamber in the photo scaled thrust chamber (at x=0.25 m). Although the advantageof using the Mach scaled demonstrator instead of the photo scaled demonstrator con-cerning the Mach number and heat load distribution is obvious, it is not clear how bothdesigns behave acoustically. Therefore, the two designs are investigated numericallywith PIANO-SAT, to predict the loss of acoustic energy through the nozzle.

3.1. Mean flow

PIANO only solves the perturbation part of the primitive quantities. This implies that thesteady state mean quantities are input to the acoustic solver PIANO-SAT. The meanflow field is calculated separately by a CFD solver in a steady state Euler or a RANScomputation. The flow field is then patched to the grid of the acoustic solver. The com-bustion process involves the oxidizer liquid oxygen and the fuel hydrogen. To simplifymatters combustion itself is not considered for the mean flow field. Instead the materialproperties are frozen to the state after combustion had taken place. At the inlet a blockprofile of the velocity normal to the inlet boundary is assumed. At the wall boundary aslip condition is applied in the case of a RANS calculation. The figures 2(a), 2(b), 2(c)and 2(d) show the results obtained with an Euler computation. Figure 2(c) and 2(d) showthe Mach number distribution on a slice through the thrust chamber. It is evident that thenozzle is supercritical, i.e. the flow reaches Mach 1 in the throat. This defines a wellsuited condition at the downstream boundary of the acoustic domain. Because acous-tic perturbations cannot travel faster than Mach 1, acoustic perturbations downstreamfrom the Mach 1 iso-surface cannot affect the acoustic field upstream inside the com-bustion chamber. Figures 2(a) and 2(b) depict the speed of sound distribution inside the

168 D. Morgenweck, F. Fassl & T. Sattelmayer

(a) Speed of sound photo scaled design (b) Speed of sound Mach scaled design

(c) Mach number photo scaled design (d) Mach number Mach scaled design

FIGURE 2. Mean flow field

thrust chamber. The maximum values for the speed of sound in the whole computationaldomain in case of the photo scaled design is 1617 m/s, whereas for the Mach scaleddesign it is 1616 m/s.

4. Decay coefficientA criterion for stability in rocket engines is the decay coefficient. Thereby the expo-

nential decay of pressure fluctuations is determined. In case no other source of acousticenergy exists and acoustic energy is not lost over other boundaries than the nozzle it-self, the decay coefficient is a characteristic quantity for the stability in rocket engines.Unlike the admittance it also accounts for convective loss of acoustic energy throughmean flow.

A first step to obtain a decay coefficient is to analyze the eigenmodes in the thrustchamber, which modes participate in the pressure oscillations. One way to find the dom-inating modes is to initially excite the oscillation in the chamber and to record the decayof the pressure perturbations. This is done by setting a Gaussian pressure pulse atsome point in the computational domain. The center of the Gaussian pulse is situatedat a some distance away from the axial axis to ensure that also transverse modes areexcited. In principle, the excitation mechanism has an influence on how strong certainmodes are initiated. Therefore, the recorded pressure signal depends on the excita-tion. However, a Gaussian pressure pulse has the ability to excite a broad band offrequencies. Thus, the main eigenfrequencies will be excited and can be observed inthe pressure signal. Figure 3(a) and 3(b) show the eigenfrequencies obtained by meansof the Fast Fourier Transform of the pressure signal recorded at 12 equally distributedmonitoring points around the first plane close to the injector head. Regarding the pres-sure signal from the other planes no additional eigenmode appears, which means thedominating eigenfrequencies can be evaluated by means of the plots shown in figure

Influence of Scaling Rules in Demonstrator Rocket Nozzles 169

0 5000 10000 150000

20

40

60

80

f [Hz]

p [P

a s]

L1

T1

T2

(a) Photo scaled design

0 5000 10000 150000

20

40

60

80

100

120

140

f [Hz]

p [P

a s]

T1

L1 T

2

(b) Mach scaled design

FIGURE 3. Eigenfrequencies

Photo scaled Mach scaled property

2620 Hz 2540 Hz longitudinal10240 Hz 10204 Hz first transverse10640 Hz 11080 Hz first transverse11630 Hz 12202 Hz first transverse16850 Hz 16817 Hz second transverse

TABLE 1. selected eigenfrequencies

3(a) and 3(b). The following table gives an overview of the eigenfrequencies selectedfor later analysis: The first longitudinal, three first transverse modes and the first sec-ond transverse mode are picked for the determination of the decay coefficient in furthercalculations. The character of the transverse modes cannot be identified uniquely. It isnot clear, whether it is a pure transverse mode or a mixed transverse-longitudinal mode.Therefore, it is only noted that the mode has a first or a second transverse fraction. Onthe other hand the character of the pure longitudinal mode can be determined easily. Inboth designs the first longitudinal mode can be identified. However, in the case of Machnumber scaling the mode is only hardly perceptible, which maybe interpreted that themode is strongly damped.

The modes identified in the eigenmode analysis were excited in a further calculationby their specific eigenfrequency. Again the Gaussian pulse is utilized, but this time asharmonic oscillating source, with the eigenfrequency of the mode. After a few cycles ofoscillation the source is switched off and the decay is recorded. The envelope of the de-caying part of the signal is obtained by a Hilbert Transformation and then approximatedby a least square fit of an exponential function.

∂te∫tb

[H(p(t)) − Ae−αt]2dt

∂α= 0 (4.1)

The α of the best fitting exponential function is the decay coefficient. The higher the

170 D. Morgenweck, F. Fassl & T. Sattelmayer

2 4 6 8 10 12x 10

−3

−200

−100

0

100

200

t [s]

p [P

a]

(a) Determination of the decay coefficient

0 0.5 1 1.5 2x 10

4

0

500

1000

1500

2000

f [Hz]

α [s

−1 ]

photomach

(b) Decay coefficients for the Mach and photoscaled thrust chamber

FIGURE 4. Decay coefficients and its determination

f [Hz] α [s−1]

2620 925.510240 263.310640 416.211630 693.716850 1018.5

f [Hz] α [s−1]

2540 1860.410204 126.111080 198.712202 384.716817 561.3

TABLE 2. Decay coefficient for left photo scaled design and right Mach scaled design

value of α the better is the damping and the less prone is the design to combustioninstabilities at this frequency. Figure 4(a) shows the pressure oscillation (blue contin-uous line) and the exponential fit (black dashed line). The decay coefficient is evalu-ated for the frequencies shown above. Figure 4(b) gives the results for the photo andMach scaled thrust chamber. The longitudinal mode for the two designs can be foundaround 2600 Hz. It is apparent that the mode is heavily damped in the case of the Machscaled design. This was already indicated in the plots of the eigenfrequencies 3(b). Thenext three eigenmodes for each design in the region between 10000 and 13000 Hzcorrespond to the eigenmodes with a first transverse mode contribution. Here, the re-sults show the opposite behavior. The Mach scaled thrust chamber is less stable thanthe photo scaled. The same can be found for the second transverse modes around16500 Hz. Furthermore, the general trend can be observed, that a higher excitation fre-quency causes more damping. This could be explained by a higher numerical dampingfor higher modes. The results are also given in table 2. On the left hand side are theresults for the photo scaled thrust chamber and on the right hand side are the results forthe Mach scaled thrust chamber.

5. Nozzle admittanceAnother way to characterize the nozzle’s impact on stability is to evaluate its admit-

tance. It provides information on how much acoustic energy is lost by acoustic propaga-

Influence of Scaling Rules in Demonstrator Rocket Nozzles 171

0 0.5 1 1.5 2x 10

4

−0.5

0

0.5

1

1.5

f [Hz]

Re(

Y)

[−]

photomach

(a) Real part

0 0.5 1 1.5 2x 10

4

−1

−0.5

0

0.5

1

Im(Y

) [−

]

f [Hz]

photomach

(b) Imaginary part

FIGURE 5. Admittances

tion through the nozzle. Furthermore, the admittance is often used in the framework ofstability analyses (e.g. by means of network models) incorporating the entire propulsionsystem to describe the influence of the nozzle. The admittance is defined as the ratio ofthe complex normal velocity and the complex pressure.

Y = ρ cmn · u

p(5.1)

The real part of the admittance indicates how much acoustic energy is lost throughthe nozzle by pure acoustic propagation. The bigger the real part is the more acousticenergy is propagated through the nozzle. The following relation can be given betweenadmittance and pure acoustic energy transport:

〈n · I〉 =1

2

|p|2ρ cm

ℜ(Y ). (5.2)

〈n · I〉 means the time averaged acoustic flux normal to a surface, which depends onlyon the amplitude of the complex pressure and the real part of the admittance. Equation5.2 reveals that the greater the real part of the admittance the more energy is lost. Forthe investigation in this section the admittance is determined directly at the entrance ofthe nozzle, i.e. at the third plane counting from the injector plane in case of the photoscaled design and at the second plane from the injector plane for the Mach scaleddesign. In general, the admittance is strongly dependent on the axial position.

The procedure for evaluating the admittance is similar to the procedure for obtain-ing the decay coefficient. The Gaussian pulse oscillates with a specific eigenfrequency.When the limit cycle is reached the complex amplitude of pressure and velocity oscil-lations are evaluated and the ratio between both is calculated. The same is done forevery eigenfrequency. Table 3 and figures 5(a) and 5(b) show the results for the realand imaginary part of the two sub-scale demonstrators. Analyzing plot 5(a) the real partof the admittance shows similar trends as the decay coefficient. The longitudinal modesaround 2600 Hz show a greater admittance than the rest. Especially for the Mach scaleddesign the admittance is very high, which implies that much acoustic energy is propa-gated through the nozzle by the first longitudinal mode. Even though it is not as clearas for the decay coefficient the modes with a first transverse contribution in the regionbetween 10000 and 13000 Hz show that the photo scaled design is again more damped

172 D. Morgenweck, F. Fassl & T. Sattelmayer

f [Hz] ℜ(Y )[−] Im(Y )[−]

2620 -0.045 0.586910240 -0.1776 -0.169910640 -0.1628 -0.070911630 -0.1214 0.223716850 -0.1816 -0.1887

f [Hz] ℜ(Y )[−] Im(Y )[−]

2540 1.2846 -0.827710204 -0.2073 -0.12711080 -0.2381 0.812412202 -0.1408 0.08516817 -0.1937 -0.1187

TABLE 3. Admittances for the photo scaled design (left) and the Mach scaled design (right)

than the Mach scaled one. The admittances for photo and Mach scaled design lie closetogether in case of the mode with second transverse contribution (frequency around16500 Hz).

It is interesting to note that both cases, the photo scaled and Mach scaled demonstra-tor, have negative real parts for all transverse modes. This implies that acoustic energyis propagating backward from the nozzle to the combustion chamber. Crocco [12] andCulick [13] talk in this case of the destabilizing character of the nozzle. Acoustic energy istransported back into the combustion chamber by pure wave transport. This argumentholds true only if pure acoustic transport is considered. In a fluid in motion additionalterms for the transport of acoustic energy occur which account for the convective trans-port. The sum of all those terms cause a time averaged positive flux of acoustic energy,meaning a loss of acoustic energy. The next section will show the relation between thedifferent terms of the acoustic energy flux.

6. Acoustic energy fluxIrrespective of the type of the mean flow a general conservation law for the energy of

the perturbation quantities can be given [14]:

∂E′

∂t+ ∇ · I ′ = S′ (6.1)

Thereby E′ denotes the energy of the fluctuating quantities, I ′ the flux of the fluctuatingquantities and S′ the source term. Depending on the complexity of the mean flow moreor less terms have to be considered for E′, I ′ and S′. In the simplest case, wherethere is no mean flow, E′ and I ′ equal the acoustic energy Ea and the acoustic flux Ia

respectively.

Ea = p′2

2c2mρ + ρu′2

2

Ia = p′u′

S′ = 0 (6.2)

The acoustic flux Ia can be converted into the form of equation 5.2. For a homentropicpotential flow E′ and I ′ can still be substituted by the acoustic energy and the acousticflux respectively, but some additional terms are present in their definition.

Ea = p′2

2c2mρ + ρu′2

2 + (u·u′)p′

c2m

Ia = p′u′ + p′2uρc2

m+ ρ(u · u′)u′ + p′

c2m

(u · u′)u

S′ = 0 (6.3)

Influence of Scaling Rules in Demonstrator Rocket Nozzles 173

Those terms also account for the convective transport. The source term is still zero.The formulation above can be applied in the analysis here. The flow field in the thrustchamber, like it is used in the above analysis, is homentropic and can indeed be treatedas potential flow, as it was shown by Candel [15]. If numerical damping is neglected theonly loss of acoustic energy occurs through the nozzle. Hence, integrating < n·Ia > overa circular plane perpendicular to the axial axis provides the complete loss of acousticenergy. The acoustic flux averaged over one period can be written in the following form:

〈n · Ia〉︸ ︷︷ ︸flux

=1

T

t+T∫

t

n ·

p′u′

︸︷︷︸term 1

+p′2u

ρc2m︸︷︷︸

term 2

+ ρ(u · u′)u′

︸ ︷︷ ︸term 3

+p′

c2m

(u · u′)u

︸ ︷︷ ︸term 4

dτ, (6.4)

where term 1-4 denote the different contributions to the overall flux. By means of theadmittance equation 6.4 can be reformulated to be only dependent on the pressureamplitude and the admittance:

〈n · Ia〉︸ ︷︷ ︸flux

=1

2

|p|2ρcm

ℜ(Y )︸ ︷︷ ︸

term 1

+ M︸︷︷︸term 2

+M |Y |2︸ ︷︷ ︸term 3

+M2ℜ(Y )︸ ︷︷ ︸term 4

. (6.5)

It becomes apparent that term 1 and 4 are linearly dependent on ℜ(Y ), term 3 dependson the square of the admittance, whereas term 2 is independent from the admittance, itonly depends on the Mach number M . In the following the contributions of the differentterms to the overall flux are evaluated. The procedure follows the one explained forobtaining the decay coefficient. The different eigenmodes will be excited by a Gaussianpulse oscillating with the eigenfrequency of the specific eigenmode. After a few cyclesthe source is switched off and the different terms of 6.4 are integrated over a window ofthe length of one cycle. The window is moved in time so that 〈n · Ia〉 (t) is still a functionof time. To be able to compare the results for different modes and designs to each otherthe different terms of 6.5 are normalized in each case with the square of the pressureamplitude of the wave traveling downstream. The results were taken at the third planecounting from the injector for the Mach scaled and the photo scaled demonstrator.

Figure 6(a) shows the contributions of the different terms in equation 6.4. The first lon-gitudinal mode is excited at a frequency of 2620 Hz of the photo scaled design. Thereare mainly three contributions to the overall acoustic flux, which stem from term 1-3.Term 4 plays only a minor role. The fluxes still show a oscillating behavior, which sug-gests that there are also other modes excited besides the pure longitudinal mode. Term2 gives the biggest contribution to the overall flux. This term is not dependent on theadmittance. On the right hand side in plot 6(b) the contributions to the acoustic energyflux for the Mach scaled design are shown. The excitation frequency is 2540 Hz. It canbe seen that the decay of the overall flux is very strong. There term 3 has the biggestimpact, but term 1 and 2 are still in the order of magnitude. Figure 6(c) compares the twooverall acoustic energy fluxes. It shows the Mach scaled design rapidly looses acousticenergy compared to the photo scaled one. The Mach scaled design starts with a biggervalue and quickly drops below the curve of the photo scaled design. These results agreewith what was figured out in the analysis of the decay coefficient (compare to table 2).The longitudinal mode is damped very strongly in the Mach scaled demonstrator.

Comparing the different terms for a first transverse mode of the photo scaled demon-

174 D. Morgenweck, F. Fassl & T. Sattelmayer

0 1 2 3x 10

−3

−1000

0

1000

2000

3000

t [s]

Pv [s

−1 ]

term 1term 2term 3term 4flux

(a) Photo scaled demonstrator excitation fre-quency 2620 Hz

0 1 2 3x 10

−3

−2000

0

2000

4000

6000

t [s]

Pv [s

−1 ]

term 1term 2term 3term 4flux

(b) Mach scaled demonstrator excitation fre-quency 2540 Hz

0 1 2 3x 10

−3

−1000

0

1000

2000

3000

4000

5000

t [s]

Pv [s

−1 ]

photo scaledmach scaled

(c) Comparison of the overall flux betweenMach and photo scaled design

FIGURE 6. Acoustic fluxes over time for L1 mode

strator, the importance of term 2 is emphasized again. The excitation frequency lies at11630 Hz. The only terms which are of greater importance are Term 1 and 2, whereasTerm 2 is positive and Term 1 is negative. This means Term 1 causes acoustic energyto travel from the nozzle into the combustion chamber. However, it can also be seen,that the convective transport is compensating the pure acoustic transport from term 1and the overall flux lies between term 1 and term 2 and is mainly pointing outwards. Thesame holds true for the Mach scaled geometry. The excitation frequency is at 12202 Hz.Term 1 and term 2 are strictly opposing each other so the resulting energy flux is verysmall, but still pointing outwards. Comparing the overall energy flux of both geometriesreveals that the photo scaled design is more damped than the Mach scaled design forthe first transverse mode picked above (see figure 7(c)). This is in accordance with whatwas found for the decay coefficient (see table 2).

7. ConclusionsIt was shown that PIANO-SAT is capable of investigating acoustic oscillations in a

rocket engine. The influence of nozzle design on acoustic damping were shown for twodifferent sub-scale demonstrators. Different characteristic quantities were presented,

Influence of Scaling Rules in Demonstrator Rocket Nozzles 175

0 0.002 0.004 0.006 0.008 0.01−4000

−2000

0

2000

4000

6000

t [s]

Pv [s

−1 ]

term 1term 2term 3term 4flux

(a) Photo scaled demonstrator excitation fre-quency 11630 Hz

0 0.002 0.004 0.006 0.008 0.01−1.5

−1

−0.5

0

0.5

1

1.5x 104

t [s]

Pv [s

−1 ]

term 1term 2term 3term 4flux

(b) Mach scaled demonstrator excitation fre-quency 12202 Hz

0 0.002 0.004 0.006 0.008 0.01−500

0

500

1000

1500

t [s]

Pv [s

−1 ]

photo scaledmach scaled

(c) Comparison of the overall flux betweenMach and photo scaled design

FIGURE 7. Acoustic fluxes over time for a T1 mode

which were used under different conditions. The admittance serves as boundary con-dition at the end of the cylindrical part for more sophisticated calculations with networktools, but it lacks the ability to describe the convective transport. The decay coefficientis a measure for the overall damping independent of the actual physical mechanism foracoustic damping. The acoustic fluxes show the relation between the different termsof acoustic flux and illustrate the transport of acoustic energy through the nozzle. Itwas found that the first longitudinal mode is heavily damped in the case of the Machscaled design, which makes the use of the Mach scaled design as sub-scale demon-strator preferable. However, the results change for the transverse modes. There theMach scaled design behaves less stable compared to the photo scaled design, but thedecay coefficients are still close to each other, so the susceptibility to combustion insta-bilities of both designs are similar. In general, it cannot be predicted, which mode willcause instability during operation. This depends also on other parameters of the com-bustion process. For instance there might be coupling between acoustics in the feedsystem and in the combustion chamber, which support either transverse or longitudinaleigenmodes. An investigation of the complete sub-scale engine is necessary to answerthe question which of the two versions of the demonstrator will behave more stable intesting.

176 D. Morgenweck, F. Fassl & T. Sattelmayer

AcknowledgmentsFinancial support has been provided by the German Research Foundation (DFG) in

the framework of the Sonderforschungsbereich/Transregio 40. We like to thank AstriumGmbH for providing us data and counsel for this work. The Leibniz–Rechenzentrum isgratefully acknowledged for providing computational resources.

References[1] CROCCO, L. AND CHENG, S.(1956) Theory of combustion instability in liquidpropellant rocket motors. Butterworth Scientific Publications.[2] MARBLE, F.E. AND CANDEL, S.M. (1977) Acoustic disturbance form gas non-uniformities convected through a nozzle. Journal of Sound and Vibration, 55, 225–243.[3] BELL, W.A. (1972) Experimental determination of three-dimensional liquid rocketnozzle admittances. Georgia Institute of Technology, Atlanta, PhD thesis.[4] ZINN, B.T. (1972) Review of nozzle damping in solid rocket instabilities.AIAA/SAE 8th Joint Propulsion Specialist Conference, New Orleans, Louisiana.[5] BELL, W.A. AND ZINN, B.T. (1973) The prediction of three-dimensional liquidrocket nozzle admittances. NASA CT-121129.[6] CANTRELL, R.H. AND HART, R.W. (1964) Interaction between sound and flow inacoustic cavities: mass, momentum and energy considerations. The Journal of theAcoustical society of America, 36(4), 697–706.[7] MORFEY, C.L. (1970) Acoustic energy in non-uniform flows. Journal of Soundand Vibration, 14(2), 159–170.[8] PIERINGER, J.E. (2008) Simulation selbserregter Verbrennungsschwingungenin Raketenschubkammern. PhD thesis, Technische Universität München.[9] DELFS, J.W., GROGGER, H.A. AND LAUKE, T.G.W. (2002) Numerical simulationof aeroacoustic noise by DLR-s aeroacoustic code PIANO. Technical Report, DLRBraunschweig.

[10] PIERINGER, J.E. AND SATTELMAYER, T. (2006) Simulation of thermo-acousticinstabilities including mean flow effects in the time domain. ICSV13, Vienna.

[11] EWERT, R. AND SCHRÖDER, W. (2003) Acoustic perturbation equations basedon flow decomposition via source filtering. Journal of Computational Physics, 144,365–398.

[12] CROCCO, L. AND SIRIGNANO, W.A. (1967) Behavior of supercritical nozzlesunder three-dimensional oscillatory conditions. North Atlantic Treaty Organization,AGARDograph 117.

[13] CULICK, F.E.C. (1988) Combustion instabilities in liquid-fueled propulsion sys-tems an overview. Combustion instabilities in liquid fueled propulsion systems,AGARD-CP 450, 1–2, 1–73.

[14] MYERS, M., K. (1991) Transport of energy by disturbances in arbitrary steadyflows. Journal of Fluid Mechanics, 226, 383–400.

[15] CANDEL, S., M. (1975) Acoutic conservation principles and an application toplane and modal propagation in nozzles and diffusers. Journal of Sound and Vibra-tion, 41(2), 207–232.