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Effects of mesh resolution on Large Eddy Simulation

of reacting flows in complex geometry combustors

G. Boudier a, L.Y.M. Gicquel a, and T.J. Poinsot b

aCERFACS, 42 Avenue G. Coriolis, 31057 Toulouse cedex, France

bInstitut de M ecanique des Fluides de Toulouse, Avenue C. Soula, 31400 Toulouse, France

Abstract

The power of present parallel computers is becoming sufficient to apply Large Eddy Simu-

lation (LES) tools to reacting flows not only in academic configurations but also in real gas

turbine chambers. The most limiting factor to perform LES of real systems is the mesh size

which directly controls the overall cost of the simulation, so that the effects of mesh resolu-

tion on LES results become a key issue. In the present work, an unstructured compressible

LES solver is used to compute the reacting flow in a domain corresponding to a sector of a

realistic helicopter chamber. Three grids ranging from 1.2 to 44 million elements are used

for LES and results are compared in terms of mean and fluctuating fields as well as of pres-

sure spectra. Results show that the mean temperature, species and velocity fields are almost

insensitive to the grid size. The RMS field of the resolved velocity is also reasonably in-

dependent of the mesh while the RMS fields of temperature exhibit more sensitivity to the

grid as expected by the fact that most of the combustion process proceeds at small scales.

The acoustic field exhibits a limited sensitivity to the mesh, suggesting that LES is adapted

to the computation of combustion instabilities in complex systems.

Key words: Mesh resolution, Industrial GT, Large Eddy Simulations

Corresponding author.Email address: lgicquel@cerfacs.fr (L.Y.M. Gicquel).

Preprint submitted to Elsevier 2 April 2008

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1 Introduction

Ongoing developments for the next generation of gas turbines focus on lean pre-

mixed operating regimes to satisfy emission regulations. The design of these com-

bustion chambers is complex because combustion concepts leading to minimum

emissions are also sensitive to combustion instabilities [13]. These instabilities

are due to a combination of the natural unstable modes of swirling flows (Precess-

ing Vortex Cores or PVCs [48]) with acoustics and unsteady heat release [912].

To study combustion instabilities but also to provide more accurate results for sta-

ble reacting flows, the best numerical technique developed today is Large Eddy

Simulation (LES) [1315]. LES has been used successfully for many academic

flames in simple geometries [1622] but still very rarely for complex realistic

combustion chambers. Multiple issues remain to be investigated before LES can

be used efficiently for design of combustion chambers: high-order schemes, Sub-

Grid Scale (SGS) tensor and flux vectors, flame/turbulence interaction, chemical

schemes, boundary conditions, parallel efficiency, etc.

In this framework, a fundamental and unresolved question is often neglected: the

effects of mesh resolution on LES results. Multiple authors have underlined the im-

portance of this point for LES [23,24]. Although LES results depend on mesh res-

olution (unlike Reynolds-Averaged Navier-Stokes simulation (RANS) which must

produce grid independent results as soon as the mesh is sufficiently refined), they

must satisfy multiple properties: time-averaged values must converge, Root Mean

Square (RMS) resolved values must increase when the mesh cell size decreases

and the SGS turbulence level diminishes, the resolved velocity spectra must fill

towards larger wave-numbers. In practice, these behaviors are expected to be con-

trolled by the LES models, the flow Reynolds number, the grid resolution as well as

the accuracy of the numerical solver (in the context of implicit filtering [23,25,26]).

Mesh dependency analysis of non-reacting LES predictions has recently been ad-

dressed [2729] and quality criteria have been proposed for a posteriori evaluation

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of the LES flow predictions [24,3033].

For reacting flows, the computer power needed to simulate realistic geometries is

so large that the grids used for LES are usually as large as possible while being still

too coarse to resolve all flow zones: in these circumstances, multiplying the num-

ber of grid points by a significant factor to verify the effects of grid resolution on

the LES results was impossible until very recent times. Very few LES of reacting

flows have been devoted to mesh dependency in simple configurations [2729] and

none of them has addressed this issue in complex geometry combustors. The situ-

ation has changed in the last two years: porting LES codes on massively parallel

machines in the Top 20 list has allowed a sudden increase of power for combus-

tion computations. For example, Fig. 1 shows the speed-up obtained on such a

parallel architecture for a 40 million cell configuration corresponding to a full gas

turbine combustion chamber with the parallel solver used in the present paper [34].

Speed-ups of nearly 95 percent as obtained here on 4096 processors allow to ad-

dress the problem of mesh dependency by performing one coarse grid simulation

with a reasonable mesh (typically 1.5 million cells) and then comparing it with an

intermediate grid simulation (8 times more cells) and finally with a fine grid

simulation (32 times more cells). In the present work, the mesh dependency of the

LES predictions is studied for a helicopter combustion chamber [35]. Results on the

coarse, intermediate and fine grids for the same regime allow a direct investigation

in terms of mean flow, RMS values, unsteady activity and acoustic mode excitation.

Although this grid dependency exercise must also be performed on simple aca-

demic geometries, using a real Rich-burn, Quick-mix, Lean-burn (RQL) com-

bustor case is an interesting test case because this configuration puts constraints on

meshes which are not found in most academic chambers where simple structured

meshes can be used. Using a real helicopter chamber guaranties that issues relevant

to industrial cases will be taken into account. However a drawback of this choice

is that very limited experimental information is available. Therefore the present pa-

per must be viewed only as a partial response to the problem of LES resolutions

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in combustion since no experimental result will be used for validation. For exten-

sive comparaisons of the present solver with experimental data, readers are referred

to previous studies [8,3638] where velocity and/or temperature fields have been

compared in various configurations.

The LES solver used for this study and the SGS models required for such a com-

parison are described in Section 2. Details on the combustion model are given in

Section 3, while the target configuration (a sector of a helicopter combustion cham-

ber) is presented in Section 4. Section 5 then discusses results obtained on the three

grids.

2 Massively parallel Large Eddy Simulations of reacting flows

LES for reactive multi-species mixtures involves the spatial filtering operation which

reduces for spatially, temporally invariant and localised filter functions [39,40], to:

f(x, t) =1

(x, t)

+

Z

(x,t) f(x, t) G(x

x) dx, (1)

where G denotes the filter function andf(x, t) is the Favre filtered value of the

variable f(x, t) [41].

In the mathematical description of compressible turbulent flows with chemical re-

actions and species transport, the primary variables are the species volumic mass

fractions(x, t), the velocity vector ui(x, t), the total energyE(x, t) es +1/2 uiui,and the density (x, t) = N=1 (x, t). Note that (x, t) is linked to the species

mass fractions Y(x, t) and mass conservation imposes for a mixture of N species:

N=1 Y(x, t) = 1.

The fluid follows the ideal gas law, p = r T and es =RT

0 Cp dTp/, where esis the mixture sensible energy, T the temperature, Cp =

N=1 Cp,Y the fluid heat

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capacity at constant pressure and r is the mixture gas constant, which varies with

composition and is obtained by r= RW

= RN=1YW

, where R = 8.314 kg m2/(s2K)

and W is the molecular weight of the species . The viscous stress tensor, the

heat diffusion vector and the species molecular transport use classical gradient ap-proaches. The fluid viscosity follows Sutherlands law, the heat diffusion coefficient

follows Fouriers law, and the species diffusion coefficients are obtained using a

constant species Schmidt number and diffusion velocity corrections for mass con-

servation [15].

The application of the filtering operation to the instantaneous set of compressible

Navier-Stokes transport equations with chemical reactions yields the LES equa-

tions, Eqs. (2)-(4), which need modelling for the system to be closed [23,42]:

uit

+

xj( ui uj) =

xj[Pi j i j i jt], (2)

Et

+

xj( E uj) =

xj[ui (Pi j i j) + qj + qjt] + T, (3)

Yk

t+

xj(

Yk

uj) =

xj[Jj,k+Jj,k

t] + k. (4)

In Eq. (2), the unresolved SGS stress tensor i jt = (uiuj ui uj) is usually ad-

dressed through the concept of SGS turbulent viscosity model and the Boussinesq

assumption [26]. The model henceforth reads (Smagorinsky [43]):

i jt 1

3kk

ti j = 2t Si j, (5)with,

Si j = 12

uixj

+ujxi

1

3

ukxk

i j. (6)

In Eq. (5) and (6), Si j is the resolved strain tensor and t is the SGS turbulentviscosity. The Smagorinsky model [43] is used here. It expresses t as:

t = (CS)2 S. (7)

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In Eq. (7), denotes the filter characteristic length and is approximated by thecubic-root of the cell volume, CS is the model constant (CS = 0.18) and S =(2

Si j

Si j)

1/2.

In Eqs. (3) and (4), the SGS species flux Jit = (uiY ui Y), and the SGS energy

flux qit = (uiE ui E), are respectively modelled by use of the species SGS turbu-

lent diffusivity Dt =t/Sct , where Sc

t is the turbulent Schmidt number (Sc

t = 0.9

for all ). The eddy diffusivity is also used along with a turbulent Prandtl number

Prt = 0.6, so that t = tCp/Prt:

Jit

=

DtW

W

X

xi

YVc

i and qi

t = t T

xi+

N

=1

Jit

hs . (8)

In Eq. (8) the mixture molecular weight W and the species molecular weight W can

be combined with the species mass fraction to yield the expression for the molar

fraction of species : X = YW/W. In expression (8), Vc

i is the diffusion correc-

tion velocity resulting from the Hirschfelder Curtis approximation [15] and T isthe modified filtered temperature which satisfies the modified filtered state equa-

tion [4446], p = rT. Finally, hs stands for the enthalpy of species . Althoughthe performance of the models could be improved through the use of a dynamic

formulation [25,44,4749], they are considered sufficient to address the present

investigation. Note that throughout the work, the variations of the molecular coef-

ficients resulting from the unresolved fluctuations are neglected so that the various

expressions for the molecular coefficients become only functions of the filtered

field [15].

3 Combustion modelling

This section presents the models necessary to take into account chemical kinet-

ics and flame/turbulence interactions. For simplicity, a one-step chemistry model,

Eq. (9), is derived for C10H16 based on a detailed model ofC10H16/Air combustion

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with 43 species and 174 steps (Turbomeca private communication).

C10H16 + 14 O2 10 CO2 + 8 H2O (9)

The reduced one-step scheme guaranties proper flame speed predictions only in

the lean regime (i.e. with equivalence ratios, < 1). For the target configuration,

such a chemical scheme is not sufficient to predict proper flame positionning since

the local equivalence ratio reaches a wide range of values. To circumvent such a

shortcoming the pre-exponential constant of the one-step scheme is adjusted versus

local equivalence ratio to reproduce the proper flame speed dependency on the rich

side [50]. The final expression for the rate of reaction of Eq. (9), is:

Q= A()

YC10H16WC10H16

n1 YO2WO2

n2exp(

TaT

) (mol.m3.s1), (10)

where n1 = 1.5,n2 = 0.55,Ta = 3608.4 K and the A() function is:

A() =3.841014

2(1 + tanh(

1.390.26

))

+0.33

4(1 + tanh(

1.60.8

))(1 + tanh(1.85

0.8)). (11)

Figure 2 shows that the adjusted one-step scheme matches the detailed scheme

reasonably well in terms of flame speed and adiabatic temperature for premixed

laminar flames at 8 bar, which is the target pressure for the full combustor.

The flame/turbulence interaction is modelled using the Dynamic Thickened Flame

(DTF) model. Following the theory of laminar premixed flames [51] the flame

speed S0

L and the flame thickness 0

L of a premixed front may be expressed as:

S0L A and 0L

S0L=

A, (12)

where is the thermal diffusivity and A the pre-exponential constant. Increasing

the thermal diffusivity by a factor F, the flame speed is kept unchanged if the pre-

exponential factor is decreased by the same factor. This operation leads to a flame

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thickness which is multiplied by F and more easily resolved on a coarser mesh.

While in reacting zones, diffusion and source terms issued from the thickened re-

action are well resolved and turbulence is solely represented by the efficiency func-

tion E [52], molecular and thermal diffusion cannot be over-estimated by a factor Fin mixing zones where no combustion occurs (it would yield over-estimated mixing

and wrong flame positions). Dynamic thickening is thus introduced to account for

these points [15,36,53,54]. The baseline idea of the Dynamically Thickened Flame

(DTF) model is to detect reaction zones using a sensor S and to thicken only these

reaction zones, leaving the rest of the flow unmodified. Thickening depends on the

local grid resolution and it locally adapts the combustion process to reach a numer-

ically resolved flame front. The flame sub-grid scale wrinkling and interactions at

the SGS level are supplied by the efficiency function [15,36,53,54].

4 LES solver and target configuration

The LES code used here solves the fully compressible, multi-species (variable heat

capacities) Navier-Stokes equations using a finite-volume discretization, on struc-tured, unstructured and hybrid meshes. Second and third-order temporal and spa-

tial numerical schemes [55,56] offer reliable unsteady solutions for complex ge-

ometries as encountered in the field of aeronautical gas turbines. In this study the

Lax-Wendroff second-order numerical scheme is chosen for the three meshes.

The configuration (Fig. 3) corresponds to a helicopter combustion chamber where

fuel is injected using an inverted cane injection system, also called pre-vaporizer.

The computational domain focuses on a 36 degree section of a full annular reverse-

flow combustion chamber designed by Turbomeca (Safran group). A premixed

gaseous mixture of C10H16 enters the chamber through the pre-vaporizer, Fig. 3

(a) & (b). Fresh gases are consumed in the primary zone, delimited by the chamber

dilution holes and the liner dome of the combustion chamber, Fig. 3 (a). To ensure

full combustion, this region of the chamber is fed with air by primary jets located

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on the inner liner, Fig. 3 (b). Burnt gases are then cooled by dilution jets or cooling

films located on the inner and outer liners as well as on the return bend of the com-

bustion chamber. Multi-perforated plates also ensure local wall cooling in areas of

the chamber shown on Fig. 3 (b).

The combustion regime expected in such burners mixes rich partially premixed

flames in the chamber primary zone (the gases injected in the canes can be con-

sidered as premixed gases at an equivalence ratio of 3.17) and diffusion flames

in the dilution region: i.e. RQL concept. A turbulent combustion model able to

handle both regimes is therefore needed and the DTF model offers this capac-

ity [36,54,57,58]. The three meshes used to assess the impact of the grid resolution

on reacting LES are shown on Fig. 4. Grid characteristics are summarized in Ta-

ble 2 in terms of number of cells, points and minimum/maximum cell volumes. The

meshes are refined in the primary zone, particularly in the lower part where combus-

tion occurs, and in the regions of cooling films. Due to the restrictive size of cooling

films, manipulations of the fluxes through the faces are needed to specify the cor-

rect Boundary Conditions. This particular approach is applied for the cooling air

effusing from the multi-perforated plates. These plates are also homogeneized:

the flow issuing from the perforations is redistributed over the entire surface and

corrections are performed for momentum fluxes and turbulent law-of-the-walls to

account for jet penetrations and wall interactions [59]. The Navier-Stokes Charac-

teristic Boundary Conditions (NSCBC) [56,60] are applied on the other inlet and

outlet boundaries to control the acoustic behavior of the system. Walls are adiabatic

and are treated with a turbulent law-of-the-wall to take into account boundary layer

effects, Table 1. Side boundaries of the computational domain are axi-periodic.

The operating point corresponds to cruising conditions and is the same for the three

grids. The initial condition for the three computations is built as follows: a statisti-

cally stabilized instantaneous solution, obtained with the coarse grid, is interpolated

on the two other meshes. These interpolated solutions are then computed until the

same physical time t0 to ensure independence on the initial guesses. Finally tem-

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poral integration is started for all three grids at t0 until statistical convergence of

the various mean quantities. The averaging time, the time steps and the local CPU

effort are summarized in Table 2. Note that the cost of the three computations for

the same physical time goes from 315 hours on the coarse mesh to 30 ,200 on thefine one (96 times more) so that knowing whether fine grids are really needed and

which additional information they provide is indeed a critical issue to choose a

mesh resolution.

5 Results and discussion

To evaluate the impact of the grid resolution on the LES predictions, the arguments

developed in Pope [24] are addressed. Indeed, it is now recognized by the LES com-

munity that two LES instantaneous fields may differ for various reasons [23,25,61].

Figures 5 and 6 illustrate the statement as three instantaneous views obtained from

the three meshes depict degrees of similarity and local differences. The questions of

interest are in this context whether these LES converge towards the same statistics

and whether these statistics are independent of the grid resolution, filter size, LES

model... In this work, grid resolution is investigated as the filter to grid size ratio is

kept fixed and equal to one. The LES models are the same for the three simulations:

the Smagorinsky closure for the SGS stress tensor and the DTF model (Section 2)

for the combustion model.

Instantaneous flow field visualizations are first presented to illustrate the main flow

features and the general contribution of grid refinement when performing reactingLES of a complex geometry. In the remaining part of the section, temporally av-

eraged fields are presented and analyzed. For that study, the following notation is

adopted to refer to the Reynolds statistical operator:

f(x,t)

T

=1

T

t0+TZ

t0

f(x, t) dt, (13)

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where T is the time interval used for temporal integration (i.e.: five flow-through

times, based on the mass flow rate and the primary zone volume). In Eq. (13), the

spatial dependency of

T

is omitted but inferred 1 . The Favre notationf(x, t) is

introduced to emphasize the fact that statistics constructed here are obtained fromFavre filtered quantities and might differ from the true turbulent statistics obtained

in raw experimental measurements for example [15,62]. Furthermore, because of

the filtering procedure introduced in LES, SGS contributions can only be retrieved

based on modeling criteria as underlined by Veynante [63] and Pope [24]. Assess-

ment of the LES resolved first moment is investigated first (Section 5.2). Then,

resolved standard deviations are probed in Section 5.3 followed by estimates of the

SGS contributions [24,32,63]. Finally acoustic fields are presented in Section 5.4

using pressure resolved standard deviations and spectra at various locations within

the computational domain.

5.1 Instantaneous flow topology and flame structure

Figure 5 compares instantaneous fields of the axial component of the velocity vec-

tor scaled by the mean velocity at the inlet of the pre-vaporizer for the three com-

putational domains. Although instantaneous fields on the different meshes are dif-

ferent, the main flow structures appear to be very similar: the spreading of the jets

exiting from the pre-vaporizer is well predicted and the high-speed zones induced

by multi-perforated plates are present. Likewise, the impact of the primary jets (bot-

tom liner wall of the main chamber) is equally well captured on all three meshes.

Nevertheless, and as expected, the fine grid computation exhibits more structures

perturbating the primary jet as well as an overall greater amount of small turbulent

structures throughout the computational domain.

A crucial requirement for the LES method when applied in such complex config-

1

Tdepends on x and Tsince the studied flow is not truly stationary. However for long

enough T,

Tcan be considered as mainly a function of space.

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urations is the right prediction of the combustion phenomenon. Modelling, which

is needed to supply proper combustion enhancement due to lack of interactions at

the unresolved scales, is paramount in that context. If improperly parameterized, a

turbulent combustion model can yield different flame positions for LES computedwith different mesh resolutions.

Figure 6 compares instantaneous fields of temperature for the three resolutions.The cutting plane goes through one of the pre-vaporizer outlets (Plane 1, Fig. 3)

and is colored by the instantaneous field of temperature scaled by the inlet mean

temperature. The observations drawn from Fig. 5 for the axial component of the

velocity field also apply to Fig. 6: the temperature fields are clearly enriched

with increasing grid resolution and the impact on the temperature levels seems

reduced.

Figure 7 shows the instantaneous fields of the reaction rate (left column) andthe equivalence ratio in the reaction zone (right column) for the three grids. This

last quantity is the equivalence ratio conditioned by an index equal to zero in

the gases where no combustion takes place and to one in the reaction zones. The

flame position is almost independent of the mesh used and, as expected, the re-

solved wrinkling of the flame front increases with the mesh resolution. These

observations confirm the proper behavior of the turbulence/combustion and the

underlying mechanisms built in the DTF model. Combustion starts close to the

lips of the canes at an equivalence ratio below the value of the one in the canes

(which approaches 3). Indeed, because of the large jet velocities in the cane outlet

regions, combustion can not be sustained immidiately at the cane exits. It allows

partial mixing of the rich premixed fuel mixture with the surrounding fresh air

coming from the upper wall multi-perforated plate (cf. Fig. 3 (b)). Combustion

then occurs for equivalence ratio values approaching 1.7. It then proceeds along

the walls of the chamber (left column). All three LES, even the coarse one, pre-

dict very similar fields of reaction rate.

Figure 8 displays the thickening factor fields (left column) along with the corre-

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sponding efficiency function fields (right column) for the three grids. These re-

sults complement the predictions of the reaction rate fields, Fig. 7 (left column).

An overall agreement is observed between the three simulations. The coarse

mesh solution exhibits less wrinkling and wider flame fronts, compensated bylarger values of the efficiency function. Contours of reaction rates coincide with

contours of thickening factor and efficiency function on the three meshes. The

grid dependency of the thickening factor accompanied by a proportional adap-

tation of the efficiency function leads to similar overall positions of the flame

front. From a statistical point of view, all simulations yield similar mean flame

locations within the combustion chamber and differences may be expected for

higher moments which needs to be determined and quantified for proper assess-

ment. These points are addressed in section 5.2.

As underlined by the right column results of Fig. 7, increasing the filtered field

content with smaller turbulent structures might also impact the mixture field to yield

different local filtered compositions. Figure 9 investigates that issue by showing the

Probability Density Function (PDF) of the instantaneous resolved equivalence ratio

in reacting zones, i.e. zones where the local reaction rate is higher than a tenth of

the mean volumetric reaction rate. Combustion mainly occurs in lean premixed

zones, and the fitted one-step chemical scheme is seen to operate suitably as very

few flame elements burn in rich conditions ( > 1). The fine mesh results allow

to identify the existence of three peaks which can be identified as follows: (1) the

peak around = 1.7 corresponds to the rich premixed flames created at the cane

outlet, (2) the peak at = 1 is due to the presence of the diffusion flamelets and

(3) the lean peak coincides with the mean chamber equivalence ratio ( = 0.33).

The differences in shapes and more specifically the presence of a secondary peak

at stoechiometry ( = 1) that is more pronounced as the grid is refined, underline

potential shift in combustion regimes. Further investigations are however needed

as these observations only stem from instantaneous flow studies which will differ

for different grid resolutions as underlined previously. Only temporal statistics are

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generally of interest to LES predictions.

The preliminary investigation of the instantaneous flow field confirms the good

quality and behavior of the predictions obtained with all three resolutions. Overall

agreement is clear and differences are only local as expected by the working con-

text of LES. Further analyses are presented below to quantify the impact of mesh

resolution on LES statistics.

5.2 Mean flow results

Spatial fields of the mean temporal statistics, Eq. (13), are good indicators of the

mesh influence on LES results. In this sub-section, mean fields are investigated for

the three grids for Plane 1, Fig. 3. Figures 10 and 11 respectively show the mean

axial component of the velocity vector and the mean temperature field scaled by

corresponding mean values at the inlet of the pre-vaporizer.

A good agreement is observed between the three predictions for both quantities,

Figs 10 and 11. The jet spreading at the cane outlet observed on the mean field of

velocity of Fig. 10 is recovered on the three meshes. High temperature pockets

are located in the same zones and fill similar volumes, assessing the similar mean

behavior of combustion on the three grids (Fig. 11).

Grid resolution has an impact in various highly localized regions and some dis-crepancies are detected, especially in the mean temperature fields of Fig. 11. The

improved mesh quality of the intermediate and fine grids makes the flow behave

differently in near-wall regions where the chamber flow interacts with the flow

issuing from cooling devices. For example, the thermal boundary layer created

by the multi-perforated plates (Zone 1 and 3 on Fig. 10 (a)) on the intermediate

mesh is thinner than the one on the coarse grid. Likewise, the penetration of the

cooling film located in the upper part of the liner dome (Zone 2 on Fig. 10 (a)) is

different on the coarse grid. Despite these discrepancies, the agreement between

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these sets of predictions underlines mesh independence of the first moments for

these calculations. Convergence in terms of mesh resolution seems to be reached

for the two finer grids at least in terms of first moments.

More quantitative studies are presented on Fig. 12 where profiles in transverse cuts

going through the cane jets and primary jets at different axial positions and for the

same quantities are displayed. Comparisons between the three sets of LES predic-

tions confirm the previous observations on the mean field maps (Figs. 10 and 11).

Axial velocity profiles, Fig. 12 (a), illustrate a slight over-estimation of the axial

velocity magnitude along the center of the cane for the coarse grid predictions.

Likewise, upper boundary layer and recirculating zones located behind the primary

jets seem over-estimated on the coarse grid. The intermediate and fine mesh pro-

files collapse onto the same curves at all stations which indicates that convergence

is reached on the intermediate mesh for that quantity. Furthermore, on the finer

meshes, velocity curves prior to the chamber dome are more diffused on the coarse

grid. Figure 12 presents temperature profiles at the same stations: all grid predic-

tions are in very good agreement and discrepancies appear only in the upper part

of the primary zone (Zones 2 and 3) where the stream coming from the top cooling

film does not penetrate the chamber as much on the finer meshes as on the coarse

grid. Results on the finer meshes predict more local mixing between the chamber

flow and fresh gases coming from the cooling devices. Monotoneous convergence

is achieved for the mean temperature profiles: the intermediate mesh predictions

are positioned in-between the coarse and fine grid profiles. At other locations, such

a statement is not always true and is explained by the complexity of the flow field

which is thus very sensitive to various flow and grid parameters.

Spatial distributions of the mean reaction rates in Plane 1 are displayed in Fig. 13.

As underlined from the instantaneous snapshots and the mean field of temperature,

combustion occurs in the region delimited by the cane outlet and chamber dome

for all meshes. Mean flame anchoring is similar on all three meshes and differences

only appear when localizing the peak reaction rate distributions. Indeed, as the grid

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resolution increases, the peak value of the mean reaction rate propagates towards

the lips of the cane injector. Convergence for this purely reacting quantity is more

difficult to assess. Such results corroborate the importance of the interaction be-

tween the various models and the local available grid resolution when conductingLES. They also underline the potential differences in composition as illustrated in

Fig. 9. Such local changes, which may impact mean statistical fields, directly im-

pact Arrhenius-like combustion models such as the DTF model.

5.3 Standard deviations and SGS statistics

One criterion for true LES stems from the observation that at a given value of the

Reynolds number (preferably very large) the total variances, that is to say the sec-

ond central moment based on the unfiltered variables, should remain independent of

the mesh resolution. The evolution with the grid resolution of the resolved standard

deviation as obtained from the LES field is henceforth of great interest. Similarly

the SGS contributions need to be evaluated for a proper assessment of the asymp-

totic behavior (if existing) of the total RMS values. The following sub-section dis-

cusses such aspects and presents results obtained from the three grid resolutions for

the resolved RMS values and turbulent kinetic energy for the resolved and residual

motions [24]. The analysis is based on the spatial evolution of the mean temporal

operator, Eq. (13).

Resolved RMS fields for the filtered axial velocity component and temperature are

shown in Figs. 14 and 15 for all meshes.

Figure 14 shows no significant difference between the three predictions: the RMSaxial velocity fields are similar and values reach approximately the same levels.

This global agreement highlights the proper action of the classical Smagorin-

sky model which results in consistent predictions regardless of the mesh reso-

lution. The SGS velocity model does not infer major changes in the resolved

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RMS velocity fields. Some local discrepancies can be observed in the wake of

the primary jet and the cane jet, especially on the coarse mesh. These differences

more likely arise from the over-estimated mean velocity field at these locations

resulting from an under-resolved mesh to allow proper use of the SGS model.Despite that shortcoming on the coarse mesh, convergence is almost reached on

the intermediate grid and for most of the computational domain.

The resolved RMS temperature, Fig. 15, exhibits larger differences than the re-solved RMS velocity when the mesh resolution evolves. Higher RMS tempera-

tures are observed on the fine grid, even if levels reached in the primary zone are

very similar between the intermediate and the fine grids. A direct observation of

the flow (Figs. 5 and 6) reveals that these RMS levels are directly due to strong

flame motions which can be captured on the finer grids and not on the coarse

one. Moreover the finer grids are able to capture more flame wrinkling, implying

higher RMS temperature levels. As mentioned before, the local mean reaction

rates are similar on all meshes (Fig. 13) because the DTF model compensates

the reduced resolved flame wrinkling on the coarse mesh by a larger sub-grid

efficiency. However, the model cannot explicitely reconstruct the resolved wrin-

kling and RMS temperatures observed in Fig. 15.

Results of Fig. 14 and 15 can be quantitatively confirmed by plotting cuts extracted

from Plane 1, as done on Fig. 16. The profiles show that the resolved RMS velocity

depends weakly on the mesh while the resolved RMS temperature grows signifi-

cantly when the mesh resolution increases. At this point, it is worth discussing why

these standard deviations behave in such different ways. The main reason has ac-

tually been mentioned many times before in the literature: the velocity field (and

its standard deviation) is mainly controlled by the large structures and even the

coarse grid is sufficient to capture the RMS levels because the smallest scales are

not modelled by the classical Smagorinsky model (it only models vortex dissipa-

tion beyond a cutting length). On the other hand, combustion is mainly a sub-grid

scale phenomenon, progressing at the smallest scales so that each increase of the

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mesh resolution will reveal more flame wrinkling and produce higher RMS temper-

ature. The best which can be expected from any flame/turbulence interaction model

is to conserve the zeroth-order moment which is the mean local reaction rate be-

cause it controls the flame position. Predicting the true RMS temperature remainsa challenge for LES of reacting flows 2 which can be tackled only with more so-

phisticated models providing a precise description of the temperature and species

PDF [21,6466] (which is not done with the DTF model).

The quality of a LES computation mostly relies on the scale separation between

the length scales of turbulence captured and the one modeled by the afforded grid

resolution. A quantitative measurement of the turbulence resolution is introduced

in Pope [24] by comparing SGS turbulent kinetic energy kSGS with the resolved

turbulent kinetic energy kresolved, which reads:

kresolved =1

2

UiUiT

UiUiT

T

(14)

The estimated quantity kSGS is derived from the following expression [23]: SGS =

CM

kSGS, where SGS = t is the SGS turbulent viscosity given by the classical

Smagorinsky model [43], CM is a constant value and is the filter cutting charac-

teristic length (equal to the cell characteristic length in this study). The resolution

criterion M is thus defined as M = kSGS/(kresolved + kSGS). M varies between 0

(equivalent to DNS where no model is needed) and 1 (corresponding to RANS

where the entire turbulent spectrum is modeled). Figure 17 depicts the spatial evo-

lution ofMas obtained from the temporally averaged LES and shows no significant

discrepancy between the three grids. Regions where M> 0.2 (corresponding to re-

gions where less than 80 % of the turbulent kinetic energy is resolved) are reduced

to the near-wall regions and the potential core of jets on both grids. This criterion

2 Note that eventhough all models for flame/turbulence interactions will exhibit the same

difficulties, some models may require less points to resolve the flame front for a given mesh

resolution. For example, the G-equation model [14] will exhibit more wrinkling than the

DTF model for the same resolution.

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thus guaranties the quality of the results obtained on these three grids, even if care

must be taken when dealing with near-wall regions on the coarse grid. Note that the

Pope criterion only quantifies the applicability of SGS modeling for the velocity

field and its extension to the conserved scalar, Z, is needed to assess the modeledto resolved contributions for mixing, which is of great interest from a combustion

point of view. In that case, MZ = ZZSGS/(ZZresolved +ZZSGS) is proposed and pre-

sented on Fig. 18. Similarly to Popes criterion, the SGS term is estimated following

the expression of [63]. For mixing, the impact of the grid resolution is observed

in the same regions. Figure 18 also underlines the reduced effect of the grid reso-

lution on MZ at the flame front, which is expected since the DTF model provides

a resolved field with no SGS contribution at that specific location, the interaction

between turbulence and combustion being represented through the efficiency func-

tion. Note in that respect that other LES combustion models are expected to behave

differently at the flame front.

5.4 Acoustic fields

The prediction of thermo-acoustic phenomena is of growing interest for industrial

purposes as gas turbine manufacturers are constrained, mostly for environmental

concerns, to design combustion devices able to operate at lean regimes, which are

known to be sometimes unstable. The contribution of LES in this framework is

essential and has already proved its capability to predict flame/acoustics interac-

tions [15,36,54,58,67]. A central question to use LES for combustion instability

studies is to know whether unsteady LES results (pressure fields for example) are

grid-independent. Since the previous section has indicated that the RMS temper-

ature is dependent on grid resolution, checking whether the pressure field is also

dependent on the mesh resolution is a logical next step.

RMS pressure fields exhibit very similar patterns on all meshes (Fig. 19) even

though the finer meshes lead to slightly higher p levels. The three simulations iden-

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tify the same acoustic activity in the combustor as shown by the pressure spectra

displayed in Fig. 20 for two probes: Probe 1 is located near one of the pre-vaporizer

outlets and Probe 2 in the primary zone (Fig. 3). LES reveals two modes around

8700 and 9300Hz which are recovered on the different meshes with reasonableaccuracy. In such complex geometries, these modes can not be associated to any

simple mode (i.e. quarter wave, etc). The only statement which can be made is that

they are clearly associated to transverse activity between the upper and lower walls

of the chamber.

The fact that all LES provide similar results for the p fields must be interpreted

with care:

The unsteady pressure field is the convolution of the source terms (due to un-steady reaction rate) and of acoustic propagation in the chamber. The acoustic

propagation takes place on long wave lengths (typically tens of centimeters at

10kH z in the burnt gases) and is probably not very sensitive to mesh changes.

The mean value of the source term has also been shown to be insensitive to the

mesh but the dependence of its unsteady component on mesh resolution, on the

other hand, is not studied here. This is obviouly the next step to take.

Predicting unsteady pressure in LES of reacting flows remains a significant chal-lenge and the present results only suggest that predicting unsteady pressure in a

real combustor with LES is actually not more difficult than predicting the mean

flame position or the velocity statistics. Other studies [54,6870] have indicated

the same trend and this optimistic outcome is probably justified only for the pre-

diction of the acoustic mode frequencies (which depend mainly on the geometry

and not much on the source terms) but not for the amplitudes of the modes.

In other words, predicting whether a combustor will oscillate and at which fre-

quency is a task which is within the possibilities of LES. However predicting the

exact amplitude of the oscillation might remain more difficult for a long time.

Indeed, this amplitude will depend not only on the mesh and on the prediction of

the unsteady source terms as well as on physical dissipation mechanisms but also

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on the dissipation of the numerical method, on the acoustic boundary conditions,

etc.

6 Conclusions

Although LES is the most promising tool to study unsteady phenomena in complex

geometry combustion chambers, few studies have addressed the question of the de-

pendence of LES results to mesh resolution in realistic configurations. This paper

considers this problem by performing three LES of the same helicopter combustion

chamber, the first one with 1,242,086 elements, the second one with 10,620,245

elements and the third one with 43,949,682 elements. Multiple comparisons be-

tween the three sets of LES results are performed. The flame position, its unsteady

behaviour and the mean flow fields (velocity, temperature, reaction rate) are shown

to be reasonably insensitive to mesh resolution. The standard deviation of the re-

solved velocity field is also shown to depend weakly on resolution, as confirmed by

the Pope criterion, showing that most of the kinetic energy spectrum is resolved on

the different grids. However, the RMS temperature is shown to increase when the

mesh resolution increases, consistently with the fact that combustion is a sub-grid

phenomenon and that its largest part is in the unresolved component. Finally, even

though the RMS temperature depends strongly on mesh resolution, the acoustic ac-

tivity is shown to be less dependent, exhibiting similar acoustic modes (frequency

and amplitudes) on all the three meshes, suggesting that LES is reasonably adapted

to the study of combustion instability in complex geometry combustors.

Acknowledgments

Numerical simulations and visualizations have been conducted on the computers of

the French National Computing Center (CINES) in Montpellier, at the Barcelona

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Supercomputing Center (BSC) located in Spain and CERFACS in-house computing

facility.

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List of Tables

1 BC types and specificities for LES. 29

2 Mesh characteristics used for the coarse, the intermediate and

fine LES meshes. 30

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TABLES

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Boundary type Acoustic properties Specified quantities Relax values

Cane inlet, dilution

and primary holesNSCBC inlet un, T, P, Y 50

Outlet NSCBC outlet P 200

Inlet films Purely reflective inlet un, T, P, Y -

Multi-perforated

plates

Purely reflective inlet

with momentum

correction

un, T, P, Y -

Adiabatic wallsPurely reflective wall

with law-of-the-wall

- -

Table 1

BC types and specificities for LES.

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Coarse mesh Intermediate mesh Fine mesh

Total number of points 230,118 1,875,835 7,661,005

Total number of cells 1,242,086 10,620,245 43,949,682

Max. cell volume [m3] 3.12671108 8.97802109 4.05748 109

Min. cell volume [m3] 1.81795 1011 8.29479 1012 1.1828 1012

Time step [s] 1.52107 0.88107 0.49107

Averaging time [ms] 10 10 10

Number of iterations during averaging 65,790 108,695 204,082

Total CPU effort for a 10ms(hours) LES 315 4,550 30,200

Table 2

Mesh characteristics used for the coarse, the intermediate and fine LES meshes.

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List of Figures

1 Typical speed-up curve as obtained for the configuration studied in

this work. 34

2 Chemical scheme validation: (a) is flame speed prediction, (b)

is adiabatic flame temperature as functions of the equivalence

ratio. The detailed scheme for C10H16 contains 43 species and 174

chemical steps. 35

3 Combustion chamber: (a) 3D view and (b) side view of the

computational domain. 36

4 Mesh resolution for (a) the coarse mesh, (b) the intermediate mesh

and (c) the fine mesh, as obtained in a transversal cut passing

through one of the canes (Plane 1). 37

5 Instantaneous axial velocity field scaled by the mean inlet velocity

and obtained on (a) the coarse grid, (b) the intermediate one and

(c) the fine one. 38

6 Instantaneous temperature field scaled by the mean inlet

temperature and obtained on (a) the coarse grid, (b) the

intermediate one and (c) the fine one. 39

7 Instantaneous fields of reaction rate (a, c and e) and equivalence

ratio conditioned by the local reaction (b, d and f) on Plane 1

obtained on the coarse grid, (a and b), the intermediate grid (c and

d) and the fine one (e and f). 40

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8 Instantaneous fields of the thickening factor (a, c and e) and the

efficiency function (b, d and f) on Plane 1 obtained on the coarse

grid (a and b), the intermediate grid (c and d) and the fine one (e

and f). 41

9 Probability density function of local equivalence ratio in reacting

zones (zones where local reaction rate is higher than a tenth of

the mean volumetric reaction rate) on (a) the coarse grid, (b) the

intermediate one and (c) the fine one. 42

10 Mean axial velocity field (scaled by the mean velocity at the

pre-vaporizer inlet) obtained on (a) the coarse grid, on (b) the

intermediate grid and on (c) the fine grid. 43

11 Mean temperature field (scaled by the mean temperature at the

pre-vaporizer inlet) obtained on (a) the coarse grid, on (b) the

intermediate grid and on (c) the fine grid. 44

12 Transverse profiles in Plane 1, Fig. 3, passing through one cane

outlet: (a) mean axial velocity profiles scaled by the pre-vaporizer

inlet velocity and (b) mean temperature profiles scaled by by the

pre-vaporizer inlet temperature. 45

13 Mean reaction rate on Plane 1 obtained on (a) the coarse grid, (b)

the intermediate one and (c) the fine one. 46

14 Fields of RMS resolved axial velocity on (a) the coarse grid, (b)

the intermediate grid and (c) the fine grid. The quantities are

non-dimensionalized by pre-vaporizer inlet conditions. 47

15 Fields of RMS resolved temperature on (a) the coarse grid, (b)

the intermediate grid and (c) the fine grid. The quantities are

non-dimensionalized by pre-vaporizer inlet conditions. 48

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16 Transverse profiles in Plane 1 (cfFig. 3), passing through one cane

outlet: (a) RMS axial velocity profiles scaled by squared inlet

velocity and (b) RMS temperature profiles scaled by squared inlet

temperature. 49

17 Spatial evolution of the temporal average of the resolution criterion

M: (a) is the coarse grid, (b) is the intermediate grid and (c) is the

fine grid. Black isoline corresponds to M= 0.2. 50

18 Spatial evolution of the temporal average of the resolution criterion

MZ: (a) is the coarse grid, (b) is the intermediate grid and (c) is the

fine grid. Black isoline corresponds to M= 0.2. 51

19 RMS pressure field obtained on (a) the coarse grid, (b) the

intermediate grid and (c) the fine grid. The quantities are

non-dimensionalized by pre-vaporizer inlet conditions. 52

20 Spectra of the fluctuating pressure temporal evolution at (a) Probe

1 and (b) Probe 2 (cf Fig. 3). 53

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FIGURES

4000

3000

2000

1000Speedup

4000300020001000

Number of processors

Ideal Speedup

40.106

cells mesh

Fig. 1. Typical speed-up curve as obtained for the configuration studied in this work.

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(a)

2.0

1.5

1.0

0.5Flamespeed[m.s

-1]

3.02.52.01.51.00.5

Equivalence ratio [-]

Detailed scheme

1-step schemeFitted 1-step scheme

(b)

2600

2400

22002000

1800Tempera

ture[K]

3.02.52.01.51.00.5

Equivalence ratio [-]

Detailed scheme

1-step schemeFitted 1-step scheme

Fig. 2. Chemical scheme validation: (a) is flame speed prediction, (b) is adiabatic flame

temperature as functions of the equivalence ratio. The detailed scheme for C10H16 contains

43 species and 174 chemical steps.

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(a)

Primary holes

Fuel

inlet

Dilution holes

Probe 2

Liner dome

Probe 1

Plane 1

(b)

Multi-perforated plates Cooling

film

Multi-perforated plates

Cooling films

Cooling film for

Nozzle Guide Vane

Primary holes

Dilution holes

Exit

Fuel

inlet

Fig. 3. Combustion chamber: (a) 3D view and (b) side view of the computational domain.

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(a)

(b)

(c)

Fig. 4. Mesh resolution for (a) the coarse mesh, (b) the intermediate mesh and (c) the fine

mesh, as obtained in a transversal cut passing through one of the canes (Plane 1).

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(a)

1.422

0.711

0

-0.711

-1.422

U/Uinlet

"[ ]

(b)

1.422

0.711

0-0.711

-1.422

U/Uinlet

"[ ]

(c)

1.422

0.711

0

-0.711

-1.422

U/Uinlet

"[ ]

Fig. 5. Instantaneous axial velocity field scaled by the mean inlet velocity and obtained on

(a) the coarse grid, (b) the intermediate one and (c) the fine one.

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(a)

4.613

3.708

2.803

1.898

1

T/Tinlet

"[ ]

(b)

4.613

3.708

2.8031.898

1

T/Tinlet

"[ ]

(c)

4.613

3.708

2.803

1.898

1

T/Tinlet

"[ ]

Fig. 6. Instantaneous temperature field scaled by the mean inlet temperature and obtained

on (a) the coarse grid, (b) the intermediate one and (c) the fine one.

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(a)

5000

3750

2500

1250

0

Reaction rate[mol.m-3.s-1]

(b)

3

2.25

1.5

0.75

0

Equivalence ratio []

(c)

5000

3750

2500

1250

0

Reaction rate [mol.m-3.s-1]

(d)

3

2.25

1.5

0.75

0

Equivalence ratio []

(e)

5000

3750

2500

1250

0

Reaction rate [mol.m-3.s-1]

(f)

3

2.25

1.5

0.75

0

Equivalence ratio []

Fig. 7. Instantaneous fields of reaction rate (a, c and e) and equivalence ratio conditioned

by the local reaction (b, d and f) on Plane 1 obtained on the coarse grid, (a and b), the

intermediate grid (c and d) and the fine one (e and f).

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(a)

35

26.5

18

9.5

1

Thickening

(b)

8

6.25

4.5

2.75

1

Efficiency

(c)

35

26.5

18

9.5

1

Thickening

(d)

8

6.25

4.5

2.75

1

Efficiency

(e)

35

26.5

18

9.51

Thickening

(f)

8

6.25

4.5

2.751

Efficiency

Fig. 8. Instantaneous fields of the thickening factor (a, c and e) and the efficiency function

(b, d and f) on Plane 1 obtained on the coarse grid (a and b), the intermediate grid (c and d)

and the fine one (e and f).

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(a)

1.6

1.2

0.8

0.4

0.0Probability

density

[]

3.02.52.01.51.00.50.0

Equivalence ratio []

(b)

1.6

1.2

0.8

0.4

0.0

Probability

density

[]

3.02.52.01.51.00.50.0

Equivalence ratio []

(c)

1.6

1.2

0.8

0.4

0.0

Probability

density

[]

3.02.52.01.51.00.50.0

Equivalence ratio []

Fig. 9. Probability density function of local equivalence ratio in reacting zones (zones where

local reaction rate is higher than a tenth of the mean volumetric reaction rate) on (a) the

coarse grid, (b) the intermediate one and (c) the fine one.

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(a)

1.422

0.711

0

-0.711

-1.422

Zone 1 Zone 2

Zone 3

U /Uinlet

"[ ]

(b)

1.422

0.711

0

-0.711

-1.422

U /Uinlet

"[ ]

(c)

1.422

0.711

0

-0.711

-1.422

U /Uinlet

"[ ]

Fig. 10. Mean axial velocity field (scaled by the mean velocity at the pre-vaporizer inlet)

obtained on (a) the coarse grid, on (b) the intermediate grid and on (c) the fine grid.

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(a)

4.464

3.598

2.732

1.866

1

Zone 1 Zone 2

Zone 3

T /Tinlet

"[ ]

(b)

4.464

3.598

2.7321.866

1

T /Tinlet

"[ ]

(c)

4.464

3.598

2.732

1.866

1

T /Tinlet

"[ ]

Fig. 11. Mean temperature field (scaled by the mean temperature at the pre-vaporizer inlet)

obtained on (a) the coarse grid, on (b) the intermediate grid and on (c) the fine grid.

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(a)

-5

0

5

Verticaldistancey/R

c

ane

[-]

-2.50.0-0.40.4 -20

Fine mesh

Intermediate mesh

Coarse mesh

(b)

4202.50.0 420

Fine mesh

Intermediate mesh

Coarse mesh

-5

0

5

Verticaldistan

cey/R

cane

[-]

Fig. 12. Transverse profiles in Plane 1, Fig. 3, passing through one cane outlet: (a) mean

axial velocity profiles scaled by the pre-vaporizer inlet velocity and (b) mean temperature

profiles scaled by by the pre-vaporizer inlet temperature.

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(a)

5000

3750

2500

1250

0

Reaction rate[mol.m-3.s-1]

(b)

5000

3750

2500

1250

0

Reaction rate [mol.m-3.s-1]

(c)

5000

3750

2500

1250

0

Reaction rate [mol.m-3.s-1]

Fig. 13. Mean reaction rate on Plane 1 obtained on (a) the coarse grid, (b) the intermediate

one and (c) the fine one.

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(a)

1.193

0.925

0.656

0.388

0.119

URMS

/Uinlet

"[ ]

(b)

1.193

0.925

0.6560.388

0.119

URMS

/Uinlet

"[ ]

(c)

1.193

0.925

0.656

0.388

0.119

URMS

/Uinlet

"[ ]

Fig. 14. Fields of RMS resolved axial velocity on (a) the coarse grid, (b) the intermediate

grid and (c) the fine grid. The quantities are non-dimensionalized by pre-vaporizer inlet

conditions.47

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(a)

1

0.775

0.55

0.225

0.1

TRMS

/Tinlet

"[ ]

(b)

1

0.775

0.550.225

0.1

TRMS

/Tinlet

"[ ]

(c)

1

0.775

0.55

0.325

0.1

TRMS

/Tinlet

"[ ]

Fig. 15. Fields of RMS resolved temperature on (a) the coarse grid, (b) the intermediate

grid and (c) the fine grid. The quantities are non-dimensionalized by pre-vaporizer inlet

conditions.48

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(a)

-5

0

5

Verticaldistancey/R

ca

ne

[-]

0.60.01.00.00.50.0

Fine mesh

Intermediate mesh

Coarse mesh

(b)

1.00.00.80.40.00.80.40.0

Fine mesh

Intermediate mesh

Coarse mesh

-5

0

5

Verticaldistan

cey/R

cane

[-]

Fig. 16. Transverse profiles in Plane 1 (cfFig. 3), passing through one cane outlet: (a) RMS

axial velocity profiles scaled by squared inlet velocity and (b) RMS temperature profiles

scaled by squared inlet temperature.

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(a)

1

0.75

0.5

0.25

0

M criterion

(b)

1

0.75

0.50.25

0

M criterion

(c)

1

0.75

0.5

0.25

0

M criterion

Fig. 17. Spatial evolution of the temporal average of the resolution criterion M: (a) is the

coarse grid, (b) is the intermediate grid and (c) is the fine grid. Black isoline corresponds

to M= 0.2.50

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(a)

1

0.75

0.5

0.25

0

Mzcriterion

(b)

1

0.75

0.50.25

0

Mzcriterion

(c)

1

0.75

0.5

0.25

0

Mzcriterion

Fig. 18. Spatial evolution of the temporal average of the resolution criterion MZ: (a) is the

coarse grid, (b) is the intermediate grid and (c) is the fine grid. Black isoline corresponds

to M= 0.2.51

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(a)

0.0658

0.051

0.0362

0.0214

0.0066

PRMS

/Pinlet

"[ ]

(b)

0.0658

0.051

0.03620.0214

0.0066

PRMS

/Pinlet

"[ ]

(c)

0.0658

0.051

0.0362

0.0214

0.0066

PRMS

/Pinlet

"[ ]

Fig. 19. RMS pressure field obtained on (a) the coarse grid, (b) the intermediate grid and

(c) the fine grid. The quantities are non-dimensionalized by pre-vaporizer inlet conditions.

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(a)

20000

15000

10000

5000

0

Amplitude[Pa.s

-1]

1600012000800040000

Frequency [Hz]

Coarse meshIntermediate meshFine mesh

(b)

30000

20000

10000

0A

mplitude[Pa.s

-1]

1600012000800040000

Frequency [Hz]

Coarse meshIntermediate meshFine mesh

Fig. 20. Spectra of the fluctuating pressure temporal evolution at (a) Probe 1 and (b) Probe

2 (cf Fig. 3).

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ADDITIONALS

1 Color version of Fig. 5: i.e. Instantaneous axial velocity field

scaled by the mean inlet velocity and obtained on (a) the coarse

grid, (b) the intermediate one and (c) the fine one. 56

2 Color version of Fig. 6: i.e. Instantaneous temperature field scaled

by the mean inlet temperature and obtained on (a) the coarse grid,

(b) the intermediate one and (c) the fine one. 57

3 Color version of Fig. 7: i.e. Instantaneous fields of reaction rate (a,

c and e) and equivalence ratio conditioned by the local reaction

(b, d and f) on Plane 1 obtained on the coarse grid, (a and b), the

intermediate grid (c and d) and the fine one (e and f). 58

4 Color version of Fig. 8: i.e. Instantaneous fields of the thickening

factor (a, c and e) and the efficiency function (b, d and f) on Plane

1 obtained on the coarse grid (a and b), the intermediate grid (c

and d) and the fine one (e and f). 59

5 Color views of the instantaneous fields of temperature on Plane 1

along with an isosurface of reaction rate (in blue) obtained on the

coarse grid (a), the intermediate grid (b) and the fine one (c). 60

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ADDITIONALS

55

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(a)

(b)

(c)

Additional 1: Color version of Fig. 5: i.e. Instantaneous axial velocity field scaled

by the mean inlet velocity and obtained on (a) the coarse grid, (b) the intermediate

one and (c) the fine one.

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(a)

(b)

(c)

Additional 2: Color version of Fig. 6: i.e. Instantaneous temperature field scaled by

the mean inlet temperature and obtained on (a) the coarse grid, (b) the intermediate

one and (c) the fine one.

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(a) (b)

(c) (d)

(e) (f)

Additional 3: Color version of Fig. 7: i.e. Instantaneous fields of reaction rate (a, c

and e) and equivalence ratio conditioned by the local reaction (b, d and f) on Plane

1 obtained on the coarse grid, (a and b), the intermediate grid (c and d) and the fine

one (e and f).

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(a) (b)

(c) (d)

(e) (f)

Additional 4: Color version of Fig. 8: i.e. Instantaneous fields of the thickening

factor (a, c and e) and the efficiency function (b, d and f) on Plane 1 obtained on

the coarse grid (a and b), the intermediate grid (c and d) and the fine one (e and f).

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(a)

(b)

(c)