large-eddy-simulation prediction of indirect combustion...
TRANSCRIPT
Special Issue – Advective disturbances in combustor thermoacoustics
Large-Eddy-simulation prediction ofindirect combustion noise in the entropywave generator experiment
Stephane Moreau1, C. Becerril2 and L.Y.M Gicquel2
Abstract
Compact and non-compact analytical solutions of the subsonic operating point of the entropy wave generator experi-
ment are compared with detailed numerical results obtained by large Eddy simulations. Two energy deposition methods
are presented to account for the experimental ignition sequence and geometry: a single-block deposition as previously
used and a delayed deposition that reproduces the experimental protocle closely. The unknown inlet acoustic reflection
coefficient is assumed to be fully reflective to be more physically consistent with the actual experimental setup. The time
delay between the activation of the heating modules must be considered to retrieve the temperature signal measured at
the vibrometer and pressure signals at the microphones. Moreover, pressure signals extracted from the large Eddy
simulations in the outlet duct using the delayed ignition model clearly reproduce the experimental signals better than the
analytical models. An additional simulation with actual temperature fluctuations directly injected at the inlet of the
computational domain clearly shows that the pressure fluctuations produced by the acceleration of the hot slug
yields indirect noise almost entirely. Finally, the entropy spot is shown to be distorted when convecting through the
turbulent flow in the entropy wave generator nozzle. Its amplitude decreases and its shape is dispersed, but hardly
any dissipation occurs. The distortion appears to be negligible through the nozzle and become important only when
convected over a long distance in the downstream duct. As the dominant frequencies of the entropy wave generator
entropy forcing are very low, the effects of dispersion by the mean flow are however weak.
Keywords
Nozzle flow, combustion noise, large Eddy simulation
Date received: 27 April 2017; accepted: 11 October 2017
1. Introduction
Modern turboengine architectures involve lean, par-tially premixed and more unstable combustors,1 alongwith fewer turbine stages to prevent the propagation ofcombustion noise outside of the engine.2 Similarly,turboshaft engines have even fewer turbine stages andlower ejection velocity and consequently no jet noise tomask the combustion noise.3,4 As a result of suchtechnological changes, the latter noise source is becom-ing a major nuisance that needs to be understood andcontrolled to meet future regulations. Two noise mech-anisms are usually evoked: a direct combustion noisecaused by the fluctuations of the flame heat release,5
and an indirect combustion noise mostly caused bythe acceleration of entropy spots,6 which could not beexperimentally evidenced until recently. Indeed an
experiment at the German Aerospace Center DLRtermed entropy wave generator (EWG) was specificallydesigned to study the latter, and was the first to con-clusively show the generation of indirect noise thateluded the community for several decades.7,8
Two EWG reference cases have been consideredsince then: a supersonic case (case 1) where the nozzlethroat is choked and a shock wave is present in the
1Departement de Genie Mecanique, Universite de Sherbrooke,
Sherbrooke, QC, Canada2Cerfacs, Toulouse, France
Corresponding author:
Stephane Moreau, Departement de Genie Mecanique, Universite de
Sherbrooke, 2500 bd de l’universite, J1K 2R1, Sherbrooke, QC, Canada.
Email: [email protected]
Creative Commons CC BY-NC: This article is distributed under the terms of the Creative Commons Attribution-NonCommercial 4.0 License (http://www.
creativecommons.org/licenses/by-nc/4.0/) which permits non-commercial use, reproduction and distribution of the work without further permission provided the
original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/open-access-at-sage).
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diverging section and a subsonic case (case 2) for whichthe Mach number at the nozzle throat MNth
is highenough to observe the saturation of indirect noise(MNth
¼ 0:7). Eventhough the supersonic case hasbeen modeled and simulated with success,9,10 all resultsconverging to a compact nozzle for which the indirectnoise dominates, the subsonic case has not been fullyunderstood yet, and no detailed turbulent simulationshave been achieved so far. Even the ratio of direct toindirect noise is still a matter of controversy: the ana-lytical predictions of Duran et al.11 suggest more directnoise from the heating device (HD) than indirect noisecreated at the nozzle throat, whereas unsteadyReynolds-Averaged Navier–Stokes (RANS) simula-tions by Muhlbauer et al.12 and more recently Lourieret al.13 suggest the opposite. Moreover only detailedturbulent simulations can adequately quantify theeffects of dissipation and dispersion of a hot slug, asdone by Morgans et al.,14 Giusti et al.,15 andHosseinalipour et al.16 in a simplified turbulent channelflow, or by Papadogiannis et al.17 and Wang et al.18 in ahigh-pressure turbine stage. Furthermore, the effects ofturbulent mixing or flow separation on indirect com-bustion noise as described by Howe19 can also be stu-died further within this specific numerical context. Allof these effects have not been studied in previous EWGworks, which are based on the coupling of RANS simu-lations with computational aeroacoustic (CAA) meth-odologies,20,21 unsteady RANS simulations,12,13 Eulersimulations, and even analytical modeling.9,11,22 Toreach the need for higher-fidelity flow predictions,Large Eddy Simulation (LES) seems to be a firstgood candidate to simulate the high Reynolds numberEWG subsonic test case, and provide further insightinto the physical phenomena of indirect combustionnoise generation and transmission.
First the EWG test facility and main data areoutlined. The analytical method to model the EWGexperiments that relies on the non-compact approachof Duran and Moreau22 is then described with a par-ticular emphasis on the improved HD model comparedwith Leyko et al’s.9 one. Preliminary results by Becerril
et al.23 have already shown its importance. Thenthe complete numerical strategy based on full 360�
configurations of the EWG setup with and withoutexact or approximate energy deposition in the compu-tational domain is presented. Conclusions are thendrawn and the limitations of the analytical approachhighlighted.
2. Entropy wave generator experiment
The complete domain of the EWG (illustrated inFigure 1) includes a settling chamber with a plateinstalled at the inlet to avoid the formation of the jetformed by the sudden expansion. In addition to theplate, a honeycomb flow straightener (hatched sectionin Figure 1) has been installed to minimize lateral vel-ocity components and obtain a straight plug flow enter-ing the nozzle. No experimental data on the inletimpedance are available and this parameter remainsunknown.
A HD is placed in the duct upstream of the nozzle.This HD is composed of six modules themselvescomposed of electric resistances, each module beingseparated from its neighbors by �xr ¼ 8 mm. Themost upstream heating module is located at xHD1
¼
�145:5mm from the nozzle throat. Note that experi-mentally a gap of 1.8mm separates the electrical resist-ances from the duct wall to prevent overheating andfusion of the wires. This implies that only part of theboundary layer and flow is heated, an effect that has tobe accounted for as accurately as possible in the LES.For the specific subsonic test case considered here, thetemperature fluctuation is no longer measured by athermocouple (local measurement) but by a vibrometer,which is a nonintrusive device that analyzes the changein the optical path length caused by the variation offlow density, and gives access to a temperature alonga line of sight, here the duct diameter. Indeed once theHD is activated, a hot spot is convected and the tem-perature fluctuation produced by this energy depositionis measured by this vibrometer located at xvib ¼�58:5mm from the nozzle throat. Therefore,
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-0.0
13 0 0
.25
y [m
]
2.1
x [m]
Figure 1. Sketch of the EWG complete geometry.
2 International Journal of Spray and Combustion Dynamics 0(0)
the temperature measured by this element can bedescribed as a spatially-averaged temperature over thediameter
Tvib ¼ Tmean ¼1
2R
Z R
y¼�R
Tð yÞdy ð1Þ
After the HD, the flow enters the converging–diverging nozzle in which the flow is strongly acceler-ated. Downstream of the nozzle, in another tube sectionwith a length of 1020mm and a diameter of 40mm, fourwall-flushed microphones measure the pressure fluctu-ations induced by the acceleration of the entropy wave.The outlet impedance of the setup has been measured.More details on the experiment can be found in Bakeet al.8
3. EWG analytical simulations
The analytical methodology proposed by Duran et al.11
is here revisited by introducing a model that takes intoaccount the entire HD and non-compact transfer func-tions of the nozzle. The model for the HD considerseach heated ring of the device as a compact element.Therefore, each ring generates its own acoustic andentropy waves. Balance equations for mass-flow rate_m, total temperature Tt, and entropy s can hence bewritten for each compact heated ring
_m0
_m
� �j,0
¼_m0
_m
� �j,1
ð2aÞ
T0tTt
� �j,0
þq0j 1þ� � 1
2M2
0
� ��1¼
T0tTt
� �j,1
ð2bÞ
s0
cp
� �j,0
þq0j ¼s0
cp
� �j,1
ð2cÞ
where the prime corresponds to fluctuations, the sub-script j indicates that the balance equation is applied tothe jth ring, and subscripts 0 and 1 stand for upstreamand downstream positions of the considered ringrespectively. q0j is the energy induced by the ring heat-ing. � and cp are the air heat capacity ratio and thespecific heat of air at constant pressure, respectively.M0 is the inlet Mach number. Equations (2a), (2b),and (2c) are then introduced in the non-compact invari-ant formulation of Duran and Moreau.22
3.1. HD model
To reproduce the shape of the temperature fluctuationrecorded by a thermocouple, while avoiding to simulatethe activation sequence, Leyko et al.9 introduced
a function � of time t using two exponentials. The stiff-ness of the rising and decreasing phases of the respect-ive functions was controlled with a single relaxationparameter noted �. In the present study, which followsthe same strategy, one relaxation coefficient is used forthe rising phase �1 and a second one for the decreasingphase �2
� tð Þ ¼
0 if t5 t0,
1� exp � t�t0�1
� �if t 2 t0; t0 þ Tp
� �1� expð�
Tp
�1Þ
h iexpð
t0�tþTp
�2Þ if t4 t0 þ Tp
8>>><>>>:
ð3Þ
where t0 is the triggering time of the HD ignitionsequence and Tp is the pulse duration or the durationof the energy deposition. As �Texp� tð Þ represents theactual shape of the temperature pulse measured bythe thermocouple (�Texp being the mean temperatureincrement induced by the HD), the assumption of split-ting the energy uniformly into nr activated rings at onceis implicitly made. Therefore, the energy delivered byeach heated ring reads
T 0HDj
T¼ � tð Þ
�Texp
nr Tð4Þ
One last parameter needs to be defined: the energyprovided by each HD q0j. This energy has beenintroduced in the above balance equations (equa-tions (2a)–2(c))) as an entropy perturbation sourceterm. With the assumption of an isobaric entropy per-turbation as done by Huet and Giauque,24 q0j ¼ s0HDj
=cpcan be expressed as a sole temperature perturbation, forwhich a delay �j can been introduced to take intoaccount the ignition sequence of the HD in the subsonictest case
q0j ¼s0HDj
cp� �HDj
¼T0HDj
Texp i!�j
� ð5Þ
where ! is the angular frequency.From this input, the entropy wave �HD generated by
the whole HD at the inlet of the nozzle is describedas the summation of the entropy waves �HDj
generatedby the individual rings, and recast into
�HD ¼Xnrj¼1
T0HDj
Texp �i!
Lj,1
c0M0� �j
� � �ð6Þ
where Lj,1 ¼ xNin� xHD1
� j� 1ð Þ�xr represents thedistance from the jth ring to the inlet of the nozzlexNin
. c0 is the speed of sound.
Moreau et al. 3
3.2. Results
The subsonic case of Bake et al.8 (case 2) is only studiedhere using the above analytical approach including theignition sequence and the number of rings of the HD asthe supersonic case (case 1) has been studied in detailswith success by Leyko et al.9 The ignition sequenceapplied in the subsonic test case activates each heatingring one after the other, starting by the one located atthe axial position xHD1
. The delay between each ringactivation corresponds to the time that one temperaturefront takes to reach the next ring. Therefore, it can besimply written for each jth ring as
�j ¼ � j� 1ð Þ�xrc0M0
ð7Þ
so equation (6) can be reduced to
�HD ¼ � tð Þ�Texp
Texp �i!
xNin� xHD1
c0M0
� � �ð8Þ
which is the same expression as the compact expres-sion for the most upstream heated ring (xHD1
).Moreover, since the HD can be considered as compactfor acoustics (as shown in the supersonic case), it canbe modeled as a compact element located at the axialposition xHD1
. The relaxation coefficients �1 and �2 arechosen to fit at best the experimental measurement ofthe vibrometer. In Figure 2, the temperature fluctu-ation modeled analytically is compared with theexperimental measurement as well as with the resultspublished in Duran et al.11 at the vibrometer positionxvib ¼ �58:5mm. Note that, at the time, Duran et al.did not study in detail the HD and located it at theaxial position xHDDuran
¼ �100mm, which induces a
time delay of 3.6ms in the signals. Temperature andpressure signals have therefore been shifted by 3.6msfor proper comparisons of the temperature fluctuationprofiles.
Pressure fluctuations issued by the improved tem-perature hot slug are shown in Figure 3 for three inletreflection coefficients Rin ¼ �1, 0, 1½ �. Comparisons aremade between the analytical compact theory in Duranet al.,11 the analytical non-compact invariant method inDuran and Moreau,22 Euler simulations in Duranet al.,11 and the experimental pressure traces.
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-20
0
20
40
60
80
100(a)
(b)
(c)
0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24
Pre
ssur
e flu
ctua
tion
[Pa]
Time [s]
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-20
0
20
40
60
80
100
0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24
Pre
ssur
e flu
ctua
tion
[Pa]
Time [s]
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-20
0
20
40
60
80
100
0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24
Pre
ssur
e flu
ctua
tion
[Pa]
Time [s]
Exp. signalInvariantsCompactDuran et al. 2013 - AVBPDuran et al. 2013 - Compact/non compact
Figure 3. Pressure traces recorded at the outlet of the EWG
(xout¼ 2100 mm) for different inlet reflection coefficients Rin and
a partially reflecting outlet reflection coefficient Rout
(Kout¼ 160 s�1). (a) Rin¼ -1. (b) Rin¼ 0. (c) Rin¼ 1.
-2
0
2
4
6
8
10
12
14
0.1 0.11 0.12 0.13
Tem
pera
ture
fluc
tuat
ion
[K]
0.2 0.21 0.22 0.23 0.24
Time [s]
Exp. signal
Analytic
Duran et al. 2013
Figure 2. Temperature fluctuation produced by the heating
device extracted at the vibrometer position xvib ¼ �58:5 mm.
Parameters of the model: �Texp ¼ 13:4 K, nr¼ 1, �1 ¼ 3:5 ms,
�2 ¼ 7 ms, t0 ¼ 0:1 s, and Tp ¼ 0:1 s.
4 International Journal of Spray and Combustion Dynamics 0(0)
Very different results are obtained and it should bestressed that for reflection coefficients different fromzero (Rin 6¼ 0), the pressure fluctuations obtained withthe non-compact transfer functions differ from the onesobtained with the compact transfer functions. ForRin ¼ �1, the nozzle is not compact at this operatingpoint, which disagrees with the results found by Duranet al.11 Yet, Duran et al.11 considered partially reflectiveinlet and outlet boundary conditions (Rin close to 1),for which compact and non-compact results are almostthe same. The results obtained by Duran et al.11 canthen be retrieved by taking their inlet and outlet relax-ation coefficients. Therefore, the large variance in theresults shows the great dependence of the generatedpressure fluctuations on the unknown inlet impedanceand on the temperature–pulse shape.
3.3. Summary
Results show that the HD may be considered as com-pact for acoustics, and when no time delay �j betweenthe heating rings is considered, the overall entropyfront is smoothed by the convective process takingplace when each front is transported from its originto the next heating ring. Two parameters are found tostrongly modify the generated noise induced by thetemperature fluctuation and the ratio of direct andindirect combustion noise: the shape of the hot slug(modulated by the relaxation parameters �1 and �2)that controls the peak pressure generated, and theinlet reflection coefficient that modifies the shape ofthe generated acoustic pressure signal. This inlet reflec-tion coefficient, linked to the relaxation chamberupstream of the inlet duct of the EWG seems to beclose to �1, and a partially reflective condition seemssufficient to better recover the experimental variation ofthe noise peak pressure at the outlet of the EWG. Yet,the two improvements brought to previous analyticalpredictions by Duran et al.11 do not help bring thepressure peaks closer to experiment, which justify thedetailed numerical investigation presented in the nextsection.
4. EWG numerical simulations
This analytical study of the EWG test cases allowed toassess the influence of different parameters in the trans-mitted entropy noise. However, the analytical results ofthe subsonic test case have not shown satisfactory com-parisons with experiment yet. Therefore, in order togain a better insight in the generation and propagationof indirect combustion noise, the subsonic test case at anozzle-throat Mach number of 0.7 has been simulatedby compressible LES of the full 360� EWG configur-ation in the following subsections.
4.1. Numerical set-up
Considering the whole EWG configuration as shown inFigure 1 without a model describing the honeycombflow straightener, big vortical structures are detachedfrom the disc and generate unwanted acoustic perturb-ations. Therefore, the complexity of the flow inside thesettling chamber, the lack of model to represent thehoneycomb, and the uncertainty issued by the corres-ponding upstream acoustic reflection coefficient,inferred to trim the settling chamber from the numer-ical domain. Four numerical meshes were created andthen used to study the indirect combustion noise gen-eration within the EWG nozzle flows. A coarse mesh(M0) was used for Euler calculations (not shown here).A medium mesh (M1) was used to reproduce theexperiment carried by Bake et al.8 with wall laws (max-imum dimensionless distance to the wall yþ of about 35and 20 on average). A finer mesh (M2) was then used tocompute nozzle transfer functions on a reduced meshwithout wall laws (maximum yþ of about 20 and 5 onaverage) and a very fine mesh (M3) intended to yieldreference wall-resolved simulations (maximum yþ
below 5 and on average 1 on all walls). M0, M1, andM3 cover the entire EWG configuration, from the inletof the upstream duct (x¼� 250mm) to the outlet ofthe downstream duct (x¼ 2100mm). Since the compu-tation of the transfer functions of the nozzle does notrequire to mesh the entire EWG configuration, thedomain considered is smaller (from x¼� 100mm tox¼ 400mm). All Navier–Stokes (NS) meshes arehybrid composed of prisms at the walls to improvethe resolution of the flow boundary layers and tetrahe-dral elements elsewhere. M0 has only �490 k tetrahe-dral cells (�145 k nodes). M1 is composed of �6M cells(�1.5M nodes) with four prism layers, M2 �22M cells(�5.4M nodes) with five prism layers, and M3 contains�300M cells (�80M nodes) with six prism layers. Thedifferent numerical domains and associated NS meshesare illustrated in Figure 4. They provide a gradualincrease of mesh density and prism layer to properlyassess the grid influence on the results compared withthe initial M0 mesh used in previous studies, whichcould not resolve any boundary layer and viscouseffects properly.
The simulations in this domain are achieved with theLES compressible solver AVBP,25 and the third-ordertwo-step Taylor–Galerkin numerical scheme TTGC,which allows to control numerical dissipation and dis-persion with eight points per wavelength. Within thishigh-order framework, the AVBP capability to com-pute acoustics in complex geometries has been demon-strated, for example by Truffin and Poinsot26; Martinet al.27; Selle et al.28 to study acoustic instabilities or byGiret et al.,29,30 Fosso Pouangue et al.31 and Salas andMoreau32 in aeroacoustic problems. For all the present
Moreau et al. 5
simulations, the sub-grid scale model is the Wall-Adapting Local Eddy-viscosity model.33
Boundary conditions are crucial for acoustic predic-tions. In the studies of thermo-acoustic instabilities incombustion chambers, the propagation of acoustic andentropy waves through the inlet and the outlet of thecombustion chamber determines the coupling of acous-tics and the flame, producing or not an unstable mode.At these inflow/outflow boundary conditions, Navier–Stokes Characteristic Boundary Conditions (NSCBC)are thus used to decompose flow variables into ingoingand outgoing waves and to minimize reflections.34 Forthe baseline flow, an NSCBC mass-flow rate has beenimposed at the inlet to agree with experimental data. Tobe able to impose a fully reflective inlet, this has beenreplaced by a total pressure NSCBC condition for theforced case.35 An inlet relaxation Kin¼ 50,000 s�1 isthen imposed.36 For both configurations, an NSCBCstatic pressure is imposed at the outlet with an outletrelaxation parameter Kout¼ 160 s�1 as was used byLeyko et al.9 to match the experimental impedance.For M1, this pressure has been raised to recover theoperating condition at the nozzle throat (MNth
¼ 0:7),which accounts for the lack of grid resolution to prop-erly capture the downstream pressure drop. For all NSsimulations, no-slip adiabatic boundary conditions(with possibly wall laws) are applied on all walls.
4.2. HD modeling
Leyko et al.,9 Muhlbauer et al.,12 and Lourier et al.13
have proposed different models to describe the HD ofthe experiment. Usually the model consists of the intro-duction of a volumetric power source term in theenergy equation. In this expression, the source term _Q
is the result of the product of one temporal shape func-tion �ðtÞ and a spatial shape function �ðxÞ (in the aboveanalytical model, only the temporal function is used).In order to take into account the gap between the heat-ing wires and the duct wall, another function ’ðrÞ isintroduced here to restrain the energy deposition to acylinder of radius Rdep. Furthermore, each heatingmodule is activated one after the other with a delaycorresponding to the convective time of the flow for adistance equal to the module-separation distances(8mm). Hence, the model of _Q proposed here todescribe the HD reads
_Qðx, r, tÞ ¼E0
nr
Xnrj¼1
�j ðxÞ � ’ðrÞ � �j ðtÞR R R 1�1
�j ðxÞ � ’ðrÞdVR10 �jðtÞdt
ð9Þ
�j ðxÞ ¼1
21þ tanh
x� xj þ Lj
d
� �tanh
xj � xþ Lj
d
� � �ð9aÞ
�j ðtÞ ¼
0 if t5 tj
1� exp �t�tj�1
� �if t 2 tj; tj þ Tp
� ��ðtj þ TpÞ exp �
t�tj�Tp
�2
� �if t4 tj þ Tp
8>>><>>>:
ð9bÞ
’ðrÞ ¼1 if r 2 0;Rdep
� �0 if r4Rdep
(ð9cÞ
where xj and tj are the position and the triggering timeof the jth heating ring respectively; E0 is the totalenergy introduced by the model; Lj is the half-length
Figure 4. Numerical grids for the baseline flow of the EWG. (Distances indicated in millimeters). (a) M1: Medium mesh. (b) M2: Fine
mesh. (c) M3: Very fine mesh.
6 International Journal of Spray and Combustion Dynamics 0(0)
of a heating ring n the axial (x) direction; �1 and �2 arethe relaxation times of the temporal function �j; and d isthe characteristic slope of the spatial function �. Here,tj is controlled by a time delay �� in the activation ofeach ring: tj ¼ t0 þ j� 1ð Þ��. Note that equation (9b) isthe same as the one used in for the analytical modeling(equation (3)), but the activation time corresponding toeach heating module tj is directly taken into account inthe expression. This model of energy deposition is gen-eric and reproduces (1) the energy deposition of Leykoet al.9 and Duran et al.11 where all the energy is depos-ited into a single cylinder that covers the entire lengthcovered by all the heating rings and (2) the model ofLourier et al.13 where the energy is distributed in sixdifferent heating modules, each zone being activated ata different instant (the delay being based on the con-vective time of the mean flow).
The energy E0 introduced in the simulation comesfrom the conservation of energy and reads
E0 ¼ _mcp�Tbulk Tp þ ð�2 � �1Þ 1� exp �Tp
�1
� �� � �ð10Þ
where the bulk temperature, velocity, pressure, andMach number are defined as
2TbulkðxÞ ¼
RRS UxðrÞTðx, r, �Þr dr d�
S Ubulkð11aÞ
Ubulk ¼
RRS UxðrÞ r dr d�
Sð11bÞ
PbulkðxÞ ¼
RRS pðx, r, �Þ r dr d�
Sð11cÞ
MbulkðxÞ ¼¼Ubulkffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�rTbulkðxÞ
p ð11dÞ
with S the duct cross section. Ux is the axial velocity.r is the air specific gas constant.
Two energy deposition shapes have been studiedhere: the one derived by Leyko et al.,9 called ‘‘Blockmodel’’ and the one proposed here termed ‘‘Delayedmodel.’’ The main differences between both energydepositions are:
Block: The energy is deposited within a unique cylinder
that overlaps the six heating wire modules activated at
the same time. The difference with Leyko et al.’s model
is that the volume of deposition is restrained by the
actual radius of the rings, Rdep¼ 13.2mm (smaller
than the duct), to account for the boundary layers pre-
sent in the LES and not in the Euler computation of
Leyko et al..
Delayed: The energy is spread over six cylinders of
length Lj¼ 1mm, and a time delay of activation
�� ¼ 0:702 ms between each cylinder is introduced to
closely reproduce the experimental ignition sequence.
According to equation (9), the values of the differentparameters of the deposition model for the two differ-ent test cases are listed in Table 1.
4.3. Reference LES
To assess the quality of the LES and their dependenceon the selected grids, bulk quantities as defined in theprevious section (equations (11a)–(11d)) are first com-pared with the isentropic quasi-1D theory. To do so, atime average followed by a surface average of the LESflow fields are calculated in the entire computationaldomain. Figure 5 shows three baseline flow axial evalu-ations obtained with the different meshes. As expected,the Euler results on M0 agree well with the isentropicquasi-1D theory for all variables. All simulations alsoexhibit the same evolution of Mach number and statictemperature through the nozzle. Noticeably, all bulkMach numbers at the nozzle throat (main target)agree with the experimental value, within the experi-mental uncertainty related to the indirect measuringtechnique. A similar behavior is also observed on thebulk velocity. Only the bulk pressure evolution is dif-ferent as the outlet pressure on M1 and M2 has beenadjusted to match the operating point on these meshesto compensate either the lack of resolution at the throator the shorter domain. No correction at the outlet hasbeen applied on M3, which uses the experimentalatmospheric pressure. Finally, the results on M3slightly differ at the throat because of the higher local
Table 1. List of parameters for each deposition
model test case.
Parameter Block Delayed
nr 1 6
E0 16.24 J 16.24 J
x0 �125.5 mm �145.5 mm
�xr 0 mm 8 mm
2Lj 40 mm 1 mm
Rdep 13.2 mm 13.2 mm
d 1 mm 1mm
t0 0.1 s 0.1 s
�� 0 s 0.702 ms
Tp 0.1 s 0.1 s
�1 3.5 ms 3.5 ms
�2 7 ms 7 ms
Moreau et al. 7
resolution and depart from the isentropic valuesbecause of the proper capture of the transition to tur-bulence and the consequent losses. Nevertheless, allgrids predict a local transition to turbulence close tothe throat and an attached flow (nonzero wall shearstress) in the diverging section as in the experiment.
The main mean values in the simulations and theexperiment are also summarized in Tables 2 and 3,respectively. The former provides the experimentaland the isentropic theoretical values. The latter gatherall simulation results. They are in overall good
agreement. In particular, the simulation on M3 yieldsvalues of total and static pressure that are the closest tothe measurements. Noticeably, the prediction of thetotal pressure at the nozzle throat that indicates thelocal losses is greatly improved on M3. Again all tar-geted nozzle-throat Mach number are within the experi-mental uncertainty. The low variability in the isentropicvs bulk profiles shown in Figure 5 as well as the goodcomparison with the experimental measurements ofTables 2 and 3 show that the operating point hasbeen correctly retrieved in all LES.
Additional quantitative comparisons can beobtained by looking at radial profiles of time- and azi-muthally-averaged flow properties along the EWGnozzle. Two representative positions are selected here,at the nozzle throat (xNth
¼ 0mm) and at the nozzle-diffuser outlet (xNout
¼ 250mm). Only the Machnumber and the temperature are shown as they are rep-resentative of the kinematics and the thermal state ofthe flow. Similar behaviors are found on velocity andpressure for instance.
Profiles of Mach number and temperature at thesetwo locations are shown for all NS simulations inFigures 6 and 7, respectively. They are found very simi-lar for all computations. Up to the nozzle all profiles ofMach number are identical. The latter then increasessignificantly up to the nozzle throat where the acceler-ation of the flow is the strongest and the boundary layerthe thinnest. Its thickness seems to be well captured in
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8(a)
(b)
(c)
-0.05 0 0.05 0.1 0.15 0.2 0.25
M [-
]ExpM0M1M2M3Isent.
260
265
270
275
280
285
290
295
-0.05 0 0.05 0.1 0.15 0.2 0.25
Tem
pera
ture
[K]
65000
70000
75000
80000
85000
90000
95000
100000
105000
110000
-0.05 0 0.05 0.1 0.15 0.2 0.25
Pre
ssur
e [P
a]
Figure 5. Bulk quantities computed from the LES and
compared with the isentropic theory. (a) Bulk Mach number. (b)
Bulk temperature. (c) Mean pressure.
Table 2. Comparison of parameters of the unperturbed flow
between the numerical simulation and the experiment—Part 1.
Parameter Experiment Isentropic
Mass–flow rate m:
(kg/h) 37 37
Total pressure pt (Pa) 105,640 108,025
Nozzle pressure pNth(Pa) 68,650 78,050
Outlet pressure po (Pa) 101,300 100,800
Uptream bulk velocity Ui (m/s) 11.39 11.33
Nozzle Mach number MNth0.7 0.7
Table 3. Comparison of parameters of the unperturbed flow
between the numerical simulation and the experiment—Part 2.
Parameter Euler (M0) LES (M1) LES(M2) LES(M3)
m:
(kg/h) 37 37 37 37
pt (Pa) 108,327 107,840 107,730 104,620
pNth(Pa) 75,762 69,100 71,900 66,640
po (Pa) 100,800 104,700 104,715 101,300
Ui (m/s) 11.31 11.34 11.34 11.67
MNth0.70 0.72 0.685 0.725
8 International Journal of Spray and Combustion Dynamics 0(0)
all three simulations, but a stronger acceleration alongthe axis is seen on M3, even though the reached Machnumber remains within the uncertainty of the experi-mental target (Figure 6(a)). Because of the adverse pres-sure gradient in the diffuser, the boundary layerthickens up to the outlet. The same final boundarylayer thickness is reached on the finest meshes M2and M3 (Figure 6(b)). Only on M1 the boundarylayer is thicker due to the grid resolution. Similar lowMach numbers of about 0.022 are reached on the cen-terline for all three simulations. The evolution of thetemperature is also similar in all simulations up tothe nozzle. A decrease of temperature is then seen upto the nozzle throat as a consequence of the above flowacceleration. The thermal boundary layer thicknessat the nozzle throat seems to be the same for all threecalculations (Figure 7(a)). Due to the stronger acceler-ation on M3, a lower temperature is reached locally.Yet the same temperature value is retrieved at the exitof the nozzle (Figure 7(b)). Moreover, the largest radialvariations of temperature and consequently of pressureare found in the nozzle-throat region because of thelarger curvature of the streamlines at this specific loca-tion. As a result, an unsteady azimuthal velocity is
induced in the diverging section, which can beexplained by the local radial equilibrium equation(already used in the experiment to compute the nozzleMach number23,37
@p
@r’ ��
u2�r
ð12Þ
This equation stands for the momentum conserva-tion in the radial direction for an axisymmetric flowwithout radial velocity, which is only strictly valid atthe throat. Yet, the radial velocity remains small in thepresent slowly-varying divergent. Note also that themean tangential velocity remains zero in the presentaxisymmetric setup. The generation of u� means thatvorticity is generated in the axial and radial directions(�x and �r, respectively) by the radial pressure gradient.Such vortices are then stretched and deformed by theflow acceleration through the nozzle and is a soundgeneration mechanism that contributes to the indirectnoise generation.
Overall, it can be concluded that all simulations cap-ture the targeted operating condition and have a very
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
0.0035
0.004
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Rad
ius
[m]
Mach [-]
0 0.002 0.004 0.006 0.008 0.01
0.012 0.014 0.016 0.018 0.02
0
0.0
05 0
.01
0.0
15 0
.02
0.0
25 0
.03
Rad
ius
[m]
Mach [-]
(a)
(b)
Figure 6. Temporal and azimuthal average of the Mach number
versus the radius. M1 (——). M2 (– – –). M3 (- - -). (a) Nozzle-
throat xNth¼ 0:0 mm. (b) Outlet of the nozzle diffuser
xNout¼ 250 mm.
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
0.0035
0.004(a)
(b)
255 260 265 270 275 280 285 290
Rad
ius
[m]
Temperature [K]
0 0.002 0.004 0.006 0.008
0.01 0.012 0.014 0.016 0.018
0.02
290
292
294
296
298
300
Rad
ius
[m]
Temperature [K]
Figure 7. Temporal and azimuthal average of the temperature
versus the radius. M1 (——). M2 (– – –). M3 (- - -). (a) Nozzle-
throat xNth¼ 0:0 mm. (b) Outlet of the nozzle diffuser
xNout¼ 250 mm.
Moreau et al. 9
similar flow development. Therefore, parametric simu-lations with various energy-deposition models havebeen performed on the medium grid M1.
4.4. Simulation results with HD
The influence of the energy deposition model is firstshown on the Mach number and the temperature inFigures 8 and 9. Similar evolutions are seen on theaxial velocity and pressure for instance. The instantan-eous profiles are extracted at two different instants(t1 ¼ 100ms and t2 ¼ 200ms) of the ‘‘Delayed model’’simulation, and compared with the mean profilesextracted from the baseline flow simulation withoutheating. t1 corresponds to the time when the depositionstarts and t2 when the energy deposition stops. All pro-files from the inlet of the nozzle (xNin
) to the nozzlethroat (xNth
) are found to be equivalent at t1 to themean profiles from the baseline flow simulation. Thisis due to the fact that the flow is mostly laminar withoutmuch perturbation before the nozzle throat. In thediverging section though, the unsteadiness introducedby the transition to turbulence and the growing
turbulent boundary layer caused by the adverse pres-sure gradient is clearly visible in the instantaneous solu-tion profiles. For instance, a large variation of theMach number instantaneous profiles at the outlet ofthe nozzle diffuser in Figure 8(b) is caused by the jetflapping. Profiles at t2 show the evolution of the differ-ent variables when the temperature is increased by theenergy deposition. The 13.4K increase of temperaturemeasured in the upstream duct is not enough to modifythe Mach number profiles significantly. However, thistemperature increase is noticeable in the pressure pro-files (pressure increase by about 20 Pa) as shown inFigure 10. A thickening of the thermal boundarylayer is also seen in the temperature profiles at thethroat (Figure 9(a)). It is worth noting that atthe chosen instants, there is no sound generationby the entropy wave: at t1 the hot slug has not reachedthe nozzle and at t2, the first temperature front hasalready traversed the nozzle generating a steady stateafter its passage (the temperature between t1 and t2reaching a constant value). Therefore, observed pres-sure fluctuations are only the result of boundary con-dition reflections and vortex sound.
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
0.0035
0.004(a)
(b)
0 0
.1 0
.2 0
.3 0
.4 0
.5 0
.6 0
.7 0
.8
Rad
ius
[m]
Mach [-]
0.002 0.004 0.006 0.008 0.01
0.012 0.014 0.016 0.018 0.02
0
0.0
05 0
.01
0.0
15 0
.02
0.0
25 0
.03
Rad
ius
[m]
Mach [-]
Figure 8. Azimuthal average of the Mach number versus radius.
Baseline flow simulation on M1 (þþþ). t1 ¼ 0:1 s—beginning of
the energy introduction (——). t2 ¼ 0:2 s—end of the energy
introduction (– – –). (a) Nozzle-throat xNth¼ 0:0 mm. (b) Outlet
of the nozzle diffuser xNout¼ 250 mm.
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
0.0035
0.004(a)
(b)
260
265
270
275
280
285
290
295
Rad
ius
[m]
Temperature [K]
0.002 0.004 0.006 0.008
0.01 0.012 0.014 0.016 0.018
0.02
292
294
296
298
300
302
304
306
308
310
Rad
ius
[m]
Temperature [K]
Figure 9. Azimuthal average of the temperature versus radius.
Baseline flow simulation on M1 (þþþ). t1 ¼ 0:1 s—beginning of
the energy introduction (——). t2 ¼ 0:2 s—end of the energy
introduction (– – –). (a) Nozzle-throat xNth¼ 0:0 mm. (b) Outlet
of the nozzle diffuser xNout¼ 250 mm.
10 International Journal of Spray and Combustion Dynamics 0(0)
The first instants of the energy deposition are com-pared with the experimental measurements. To do so,only the first 35ms of the energy deposition are shownin Figure 11, where the mean temperature and pressuretraces obtained in the numerical simulations are com-pared with the analytical results (shown above for theinvariants with Rin ¼ �1 in Figure 3(a)) and the experi-mental data. As already highlighted in the analyticalevaluation of the EWG, the simultaneous activationof all the heating rings (block deposition model)induces a time difference in the temperature and pres-sure signals of about 3ms due to the convection time ofthe temperature front between each heating ring. Thistime delay is taken into account in the delayed model,and is computed using the values reported by Bakeet al.,8 namely the bulk velocity in the upstream ductUbulk � 11:4 m/s and the spacing between each heatingring �xR ¼ 8 mm. A total time delay is found to beabout 3.5ms. Figure 11(a) shows the temperature evo-lution at the vibrometer position. The two delayedenergy deposition models show a very good agreementwith the experimental signal. Therefore, the parameterschosen to describe the hot slug shape with the delayedmodel reproduce the shape of the temperature fluctu-ation measured by the vibrometer well, and the delay ofactivation of each heating module as well as the rightconvection velocity of the hot slug provides the correctoperating point (the hot slug arrives at the correct timeat the vibrometer position). This also agrees with thenumerical results of Lourier et al.13 that computed thetime difference of different noise sources (direct andindirect noise). As Lourier et al.13 has already found,the first direct noise signal (noise produced by the fluc-tuating heat release of the HD) starts at about 4msafter the triggering of the energy deposition and thefirst indirect noise signal (due to the acceleration ofthe hot spot through the nozzle) arrives at about12.5ms. The present pressure fluctuations generatedby the heating and convection of the hot slug throughthe nozzle are shown in Figure 11(b) at the fourthmicrophone position (xmic4 ¼ 1150:5 mm), where timedelays computed by Lourier et al. are also presented.The HD is found to be compact for the acoustic wavesof interest and direct noise signals should then arrive atthe same time for both deposition models (t � 103:5ms). Indirect noise contribution is seen to start earlierfor the block deposition model (about 3.5ms earlier)compared with the delayed model. Therefore, betweent � 103:5 and t � 109:1 ms, only direct noise is mea-sured by the microphone. Because of the short simula-tion time and the consequent small window of time forpurely direct noise contribution, one cannot concludewhich deposition model produces more or less directnoise. Nevertheless, the overall noise produced by theblock deposition model is clearly more spread and
0
2
4
6
8
10
12
14(a)
(b)
0.1 0.11 0.12 0.13Tem
pera
ture
fluc
tuat
ion
[K]
Time [s]
Exp. signalBlockDelayedAnalytic
-20
-10
0
10
20
30
40
50
0.1 0.11 0.12 0.13
1st directnoise signal
1st indirectnoise signal(delayed)
1st indirect noise signal(Block)
Pre
ssur
e flu
ctua
tion
[Pa]
Time [s]
Figure 11. LES with heating device. Temperature and pressure
traces generated by the introduction of a hot slug by the inlet
boundary condition. (a) Temperature fluctuation at the vibrom-
eter position (xvib¼� 58.5 mm). (b) Pressure fluctuation at the
fourth microphone xmic4¼ 1150:5 mm).
0.002 0.004 0.006 0.008
0.01 0.012 0.014 0.016 0.018
0.02
104
700
104
705
104
710
104
715
104
720
104
725
104
730
Rad
ius
[m]
Pressure [Pa]
Figure 10. Azimuthal average of the static pressure versus
radius at the outlet of the nozzle diffuser (xNout¼ 250 mm).
Baseline flow simulation on M1 (þþþ). t1 ¼ 0:1 s—beginning of
the energy introduction (——). t2 ¼ 0:2 s—end of the energy
introduction (– – –).
Moreau et al. 11
reaches a lower peak value (same amount of energyintroduced by both models). Furthermore, when com-paring the numerical delayed-model pressure signalwith the analytical one, the shape of both signals isvery similar (Figure 11(b)) and the starting point ofindirect noise signals matches very well (slightly afterLourier et al.13 prediction). In the zone where directand indirect noise interact together, the analyticalmodel appears to underestimate the pressure signal byhalf when LES seems to overestimate only the peakvalue, obtaining a better agreement with the experimen-tal measurement. The differences between the analyticaland numerical pressure signals can be mostly attributedto the turbulent vortices produced in the diffuser andthe consequent produced vortex sound, and the pos-sible excitation of the nozzle jet by the entropydisturbance.
4.5. Simulation results without HD
Another LES has been achieved on M1, in which atemperature fluctuation with the same characteristicsas the ones generated by the HD is introduced in thedomain without the generated direct noise. To estimatethe amount of indirect noise generated in the LES, aplane next to the HD (x¼� 100mm) is extracted fromthe forced simulation discussed above to obtain a 2Dtemperature field that depends on time only, and that isimposed at the inlet boundary condition. Figure 12compares the result of this simulation with the delayeddeposition model and the experimental results. Thetemperature and pressure fluctuations without HDhave been shifted by 10ms (convection time of theflow to reach the HD) to make a fair comparison.The temperature fluctuations show an attenuation ofthe maximum value caused by the longer path followedby the hot slug to reach the nozzle. Despite the smallermaximum temperature fluctuation, pressure traces indi-cate that the amount of indirect noise generated isalmost the same as the overall contribution of bothnoise sources, indicating that direct noise has a verysmall contribution and that indirect noise generationis the dominant source in this experiment as previouslyfound by Lourier et al.13
4.6. Deformation of the entropy spot
As mentioned above, the bulk temperature given byequation (11a) is energy conservative by constructionand should therefore be considered to verify if theentropy fluctuations only suffer from dispersion andnot dissipation through the EWG nozzle. Such a prop-erty has already been verified by Morgans et al.14 andGiusti et al.15 for constant section turbulent channelsflows. To look at the time evolution of the temperature
fluctuations, profiles of TdimlessðxÞ ¼TbulkðxÞ� �T0,bulkðxÞð Þ
�T0,bulkðxÞ,
where �T0,bulkðxÞ is the time-averaged bulk temperatureof the baseline flow, are extracted at different axial loca-tions of the domain. Figure 13 shows the evolution ofTdimless at the nozzle outlet and x¼ 1m from the nozzlethroat for the two LES and the analytical temperatureconvection model. The latter is deduced from the 1DEuler energy equation for an adiabatic flow along thenozzle axis, and is given by the system
@Tðt, xÞ@t þ uðxÞ @Tðt, xÞ@x ¼
1�ðxÞcp
uðxÞ @pðxÞ@x þ_Qðt, xÞ
h iTðt ¼ 0,xÞ ¼ TðxÞ
(ð13Þ
where u(x), �ðxÞ, T(x), and @pðxÞ@x are extracted from the
isentropic mean flow. The shape of the temperaturefluctuation is conserved through the nozzle and itsshape is only distorted and attenuated in the down-stream duct. The entropy hot slug needs almost onemeter for its amplitude to be decreased by the effectsof the mean flow. It is important to remember that thetemperature fluctuation generated by the EWG HD is
0
2
4
6
8
10
12
14(a)
(b)
0.1 0.11 0.12 0.13Tem
pera
ture
fluc
tuat
ion
[K]
Time [s]
Exp. signalDelayedIndirect noise
-20
-10
0
10
20
30
40
50
0.1 0.11 0.12 0.13
1st directnoise signal
1st indirectnoise signal(delayed)
Pre
ssur
e flu
ctua
tion
[Pa]
Time [s]
Figure 12. LES without heating device. Temperature and
pressure traces generated by the introduction of a hot slug by the
inlet boundary condition. (a) Temperature fluctuation at the
vibrometer position (xvib¼� 58.5 mm). (b) Pressure fluctuation
at the fourth microphone xmic4¼ 1150:5 mm).
12 International Journal of Spray and Combustion Dynamics 0(0)
composed of almost only very low frequencies.8 Eventhough additional 3D effects are present in the EWGnozzle, this behavior is therefore consistent with theresult of Giusti et al.15 which noticed in a straightduct that the smaller the frequency is, the smaller thedispersion of the entropy perturbation is. In order toestimate the dissipation of the entropy wave through
the nozzle, the temporal relative integral (to the vib-rometer position) of each extracted position is com-puted and showed in Figure 14, which clearly showsthat the entropy fluctuation is only convected throughthe nozzle without dissipation.
5. Conclusion
The subsonic operating point of the EWG experimenthas been computed by LES. Two energy depositionshapes have been compared to account for the experi-mental ignition sequence and geometry: the one used byLeyko et al.,9 and the delayed deposition proposed inthis work. The unknown inlet acoustic reflection coef-ficient is assumed to be fully reflective (most physicalchoice) to have a fair comparison with the analyticalmodels for a compact nozzle of Duran et al.11 and for anon-compact nozzle of Duran and Moreau.22 The timedelay between the activation of the heating modulesmust be considered to retrieve the correct time delayin the temperature signal measured at the vibrometerand pressure signals at the microphones. Furthermore,pressure signal extracted from the LES at the fourthmicrophone using the delayed ignition model clearlyreproduces the experimental signal well, while the ana-lytical analysis did not. Moreover, the temperature fluc-tuations generated by the delayed ignition model havebeen introduced in another simulation, which clearlyshows that the pressure fluctuations produced by theacceleration of the hot slug yields indirect noisealmost entirely as was found by Lourier et al.13
Finally, the convection of the hot spot has also beenstudied, and the detailed turbulent numerical results havebeen compared with a quasi-1D convection solution of ahot spot, by looking at one dimensional fields of the tem-perature fluctuations at different instants and locations inthe nozzle. As was previously found by Morgans et al.14
in a direct numerical simulation of a temperature pulseconvecting through a turbulent channel, the amplitudeand the shape of the entropy spot gets distorted whenconvecting through the EWG nozzle (the amplitude ofthe entropy spot decreases and its shape is dispersed),but hardly any dissipation occurs. The attenuation anddistortion of the entropy spot in the simulation appear tobe negligible through the nozzle and become importantonly when convected over a long distance in the down-stream duct.Moreover since in this EWG experiment, thedominant frequencies of the entropy forcing are very low,the effects of dispersion by the mean flow are weak.
Acknowledgments
The authors gratefully acknowledge all RECORD (Researchon Core Noise Reduction) partners for sharing experimental
and numerical data, and the funding from the EuropeanUnion Seventh Framework Collaborative project RECORD
0
0.02
0.04
0.06
0.08
0.1(a)
(b)
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
ΔTbu
lk/T
0,bu
lk [-
]
time [s]
AnalyticBlockDelayed
0
0.02
0.04
0.06
0.08
0.1
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
ΔTbu
lk/T
0,bu
lk [-
]
time [s]
AnalyticBlockDelayed
Figure 13. Azimuthal average of the dimensionless bulk tem-
perature Tdimless ¼ Tbulk � T0,bulkð Þ=T0,bulk at the different positions
of the nozzle against time. (a) Outlet of the nozzle,
xNout¼ 0:25 m. (b) x¼ 1 m.
0
0.2
0.4
0.6
0.8
1
1.2
-0.2 0 0.2 0.4 0.6 0.8 1 1.2
Δ Tbu
lk/T
0,bu
lk [-
]
x [m]
AnalyticBlockDelayed
Figure 14. Relative magnitude of Tdimless (compared to its value
taken at the vibrometer position) as it is convected through the
EWG nozzle.
Moreau et al. 13
under Grant No. RG66913. The authors are specially thank-ful to Dr. Friedrich Bake from DLR for providing all theexperimental data and Frederique Duvinage from
AllianTech for the complementary informations on transferfunctions of the microphones that yielded the experimentalunsteady pressure traces.
Declaration of Conflicting Interests
The author(s) declared no potential conflicts of interest withrespect to the research, authorship, and/or publication of thisarticle.
Funding
The author(s) received funding from the European UnionSeventh Framework Collaborative project RECORD underGrant No. RG66913.
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