Lagrangian and Eulerian simulations of

evaporating fuel spray in an aeronautical

multipoint injector

F. Jaegle* B. Cuenot* T. Poinsot ‡

∗CERFACS, Av. Gaspard Coriolis31057 Toulouse, Cedex 01, France

‡Institut de mecanique des fluides de Toulouse(CNRS-INPT-UPS)

Allee du Professeur-Camille-Soula, 31400 Toulouse, France

Abstract

In an effort to reduce NOX emissions of aeronautical engines, stagedinjection systems that allow optimizing lean combustion processes for dif-ferent operating points are intensely studied at present. These systemsoften employ multipoint injection systems where a series of fuel jets isinjected perpendicularly to the airflow and subsequently undergoes at-omization and mixing. The present work evaluates the capability of twodifferent numerical strategies to simulate multipoint injectors. The eval-uation is carried out on a single injector mounted in an experimental rigwith optical access. The non-reactive operating condition corresponds toa partial load regime at 4.37 bar and 473 K in the chamber. This allowsto study droplet dynamics and evaporating processes under realistic con-ditions. Measurement data on gaseous and droplet velocities as well asdroplet diameter are available.The gas phase is simulated using the large eddy simulation (LES) solverAVBP, which has repetedly demonstrated its ability to predict unsteadyflows in complex geometries. This gaseous solver is coupled with two dif-ferent simulation approaches for the liquid phase. The first is a Lagrangianmethod, the second a mesoscopic Eulerian approach. These methods bothrely on identical models for drag and evaporation. Two-way coupling be-tween the gaseous and the dispersed phase is taken into account.The simulation cases considered include a purely gaseous flow, an Eulerianspray simulation as well as a monodisperse and a polydisperse Lagrangiansimulation. The analysis of the results focuses on validation against ex-perimental data but also on the comparison of the performance betweenthe two numerical approaches under identical conditions.

1

1 Introduction

Large Eddy Simulation has proven to be a powerful numerical tool for thesimulation of industrial-scale combustion applications. As a great number ofcombustion systems rely on the direct injection of liquid fuel, the prediction oftwo-phase flow phenomena are a crucial ingredient for such simulations. Threemajor classes of approaches to simulate such problems are widely studied: Thefirst is the direct simulation of the gas-liquid interface using level-set [28], volumeof fluid (VOF) [10], ghost fluid methods [8]. This approach allows in particularto simulate primary breakup processes. In terms of computational cost, it is,however, out of reach for industrial-scale applications.The second, very polular approach is limited to the representation of a dispersedphase, i.e. a set of droplets, which are tracked individually. This representationof a spray is combined with a classical Eulerian approach for the gaseous phase,which includes the exchange of coupling terms in both directions. This methodis referred to as the Eulerian-Lagrangian (EL) approach.The third method assumes that a spray can be viewed as a continuum, for whichtransport equations can be formulated and solved numerically similarly to theones of the gaseous phase. As the Eulerian point of view is taken for bothphases, it is known as the Eulerian-Eulerian (EE) approach.

The objective of the present work is a comparison of the EL and the EEapproaches in a realistic application case, characterized by pressurized and pre-heated (and thus evaporating) conditions in a complex, industrial-scale geom-etry. Published studies of such comparisons between EL and EE exist but areoften limited to academical cases like homogeneous isotropic turbulence, channelflows or generic configurations like bluff-body cases. Furthermore, such compar-isons often rely on different codes for EL and EE, which means that the gaseousflow can be influenced by different grids, numerical schemes etc. In the presentstudy, the solver for the gaseous phase is identical for EL and EE, the models(evaporation and drag) for the liquid phase are equivalent and differ only bytheir implementation and also the grid is the same for all cases considered. Tothe authors knowledge, comparable studies of industrial-scale applications arelimited to the work of Senoner et al. [24].

In a first part, the injector and the test configuration are described. The nu-merical framewirk is laid out, including the governing equations of the gaseousphase and the liquid phase in a Lagrangian and a Eulerian formulation. In asecond part, results from a simulation of the purely gaseous flow are described,followed by the analysis of the evaporating two-phase flow. The analysis com-prises the description of spray dynamics and evaporation processes with a strongfocus on the comparison between both approaches. For both the purely gaseousand two-phase flow, experimental data are available and allow to asses the ac-curacy of the obtained results.

2

1.1 Geometry

The aeronautical injector considered here is an example of a staged architecturethat is being studied by many manufacturers for their latest or future combus-tor concepts. The primary objective is the reduction of NOX emissions, whichis achieved by lean burning in the primary zone of the combustion chamber.In particular in partial load regimes, this may lead to issues of combustion in-stabilities or flame blowoff, which can be effectively countered by fuel staging.This method allows to optimize the combustor for stable burning conditions atdifferent operating points.

An isolated, cut-away view of the injector is shown in the left half of figure1. The two stages can be distinguished from the downstream side as two conical“bowls”, where the central pilot bowl is nested inside the main stage bowl.

Pilot stage bowl!

Main stage bowl!

Fuel

admission!

Pilot injector!

24 multipoint

injectors!

Main stage!

swirler channels!

Counter-rotating

pilot stage !

swirler channels!

Figure 1: Staged premixing swirler. Left: cut-away view with highlighted injec-tion system. Right: transparent view with highlighted swirler channels.

The right hand side of figure 1 presents a transparent view in which thethree swirler stages are highlighted. Two are of radial type, both leading intothe pilot bowl near the center of the configuration. While the innermost onedischarges into the pilot bowl at its upstream end, the flow from the secondjoins it trough a circular slot in the conical wall just before the pilot flow exitsinto the chamber. The third and main swirler is of radial type, slightly inclinedrelative to the central axis. All three swirler stages are counter-rotating relativeto each one’s neighbour, which promotes turbulent mixing in the areas wherethe flows join.

The bulk of the airflow through the injector (approx. 90 %) passes themain swirler stages. The remaining 10 % is divided between the innermostpilot swirler (3 %) and the outer pilot swirler (7%). The experimental rigconsidered here consists of a block-shaped chamber with a square cross-sectionwith the injector mounted on the upstream end and the airflow exiting thechamber through a supersonic nozzle at the downstream end (see figure 2). Theflat, lateral surfaces allow for easy optical access. The airflow is fed to the

3

Plenum!

Chamber!

Staged premixing swirler!

Outlet nozzle!

Figure 2: TLC configuration ONERA Fauga-Mauzac

chamber from a plenum through the injector and a film that is visible in figure3. The computational domain comprises a certain portion of the plenum andends shortly after the sonic throat of the exit nozzle.

2 Governing equations

2.1 Gaseous phase

The filtered momentum, total energy and species conservation equations for thegas phase are:

∂ρ ui∂t

+∂

∂xj(ρ ui uj) = − ∂

∂xj[P δij − τij − τijt] + S

l−gM,i (1)

∂ρ E

∂t+

∂

∂xj(ρ E uj) = − ∂

∂xj[ui (P δij − τij) + qj + qj

t] + Sl−gE (2)

∂ρ Yk∂t

+∂

∂xj(ρ Yk uj) = − ∂

∂xj[Jj,k + Jj,k

t] + S

l−gF (3)

where a repeated index implies summation over this index (Einstein’s ruleof summation). The index k refers to the kth species and does not follow thesummation rule.The symbols ρ, ui, E and ρk denote respectively the density, the velocity vec-tor, the total energy per unit mass and the density of the chemical species k:ρk = ρYk for k = 1 to N (where N is the total number of species). Furthermore,P denotes the pressure, τij the stress tensor, qi the heat flux vector and Jj,k thevector of the diffusive flux of species k.

4

The superscript t denotes subgrid-scale terms for the shear stress τijt, heatflux qjt and species diffusion Jj,k

t.

There are several source terms on the right-hand side that account for thecoupling from the liquid phase to the gas: Sl−gM,i = Γul − Fd,i in equation 1is the vector of momentum source terms with Fd,i being the force per unitvolume the dispersed liquid phase excerts on the gas and Γ the mass transferrate from droplet evaporation per unit volume. The source term in the totalenergy equation (eq. 2) takes the form: Sl−gE = Π + Γ 1/2u2

l − ul,iFd,i whereΠ is the heat transfer rate per unit volume. The source term in the speciestransport equations (eq. 3) is applied to the equation of the evaporating (orfuel-) species F only and equal to zero for all other species k 6= F . It has theform Sl−gF = Γδk,F . The terms Fd,i, Γ and Π take different forms for EL andEE and are given in the following paragraphs.

2.2 Liquid phase

The numerical solvers for the liquid phase are based on the assumption of adilute spray, composed of spherical droplets with negligible interactions such ascollisions or coalescence. Furthermore, the ratio between the densities of the liq-uid an the gaseous phase is considered to be large (ρl/ρg >> 1), the droplets aresupposed smaller than the LES filter width and effects of shear on the dropletsas well as secondary breakup are neglected.Both numerical solvers include the same models for droplet drag and evapora-tion. Furthermore, both solvers include full two-way coupling. In the followingsections, the governing equations of both approaches are detailed. Gaseousquantities, identified by the index ‘g’ correspond to the filtered quantities ofthe LES.

2.2.1 Euler-Lagrange

In the Euler-Lagrange (EL) approach, each droplet is tracked individually, lead-ing to a very compact set of equations. The relation between the evolution of adroplet’s spatial coordinate xp,i and its velocity vector up,i is:

dxp,idt

= up,i (4)

The droplet acceleration is obtained by a simplification of the Basset-Boussinesq-Oseen equation [5]:

dup,idt

=1τp

(ug@p,i − up,i) (5)

where ug@p,i is the gas-phase velocity interpolated at the droplet location, gi thegravitational acceleration and τp the droplet relaxation time scale, that takes

5

the form:τp =

43ρpρg

dpCD (Rep) |ug@p,i − up,i|

(6)

where ρl and ρg are the droplet and gas-phase densities, dp the droplet diame-ter, Rep the droplet Reynolds number and CD the droplet drag coefficient. It isbased on Stokes’ law for particle drag [26] with the Reynolds number correctionproposed by Schiller and Naumann [22]. The (filtered) gaseous velocity ug,i isinterpolated at the position of the particle, which is noted as ug@p,i. In the ELapproach, the direct effect of subgrid-scale fluctuations on the particle motionis neglected. This effect becomes significant when the droplet Stokes numberbased on the Kolmogorov time scale τη approaches unity [7, 16]. However, Apteet al. [2] showed that the direct effect was small for swirling separated flows withsubgrid scale energy contents much smaller than those of the resolved scales.

In an evaporating spray, the temporal evolution of the droplet mass is givenby:

dmp

dt= −πdp Sh [ρDF ] ln (1 +BM ) (7)

where Sh is the Sherwood number, which is calculated using the formula pro-posed by Ranz and Marshall [20]. The term [ρDF ] contains the diffusion co-efficient of the fuel species, DF and the density of the mixture ρ. It can beexpressed by the Schmidt number of the fuel species F and the viscosity µ as[ρDF ] = µ/ScF . The Spalding number for mass transfer, BM is expressed as afunction of the fuel vapour mass fractions at the droplet surface (index ‘ζ’) andthe droplet far-field (index ‘∞’):

BM =YF,ζ − YF,∞

1− YF,ζ(8)

The evolution of droplet temperature is expressed as:

d Tpd t

=1

mp Cp,lλπdpNu (Tp − Tg@p)

ln(BT + 1)BT

(9)

with the Nusselt number Nu corrected by the Ranz-Marshall formula [20]. Theheat conductivity in the gas phase is obtained from λ = µCP /Pr, where CP isthe heat capacity at constant pressure of the gaseous mixture and and Pr thePrandtl number. The Spalding number for heat transfer, BT can be related toBM the expression

BT = (1 +BM )Sh

Nu LeF − 1 (10)

which contains the Sherwood and Nusselt numbers as well as the Lewis numberof the fuel species, LeF .

The two-way coupling terms are calculated as discrete quantities for eachdroplet and must be translated into volumetric source terms. To do so, particlecontributions and summed inside the ensemble of the cells Dj surrounding a

6

given node j of the Eulerian grid for the gaseous phase. The volume associatedto a node j is noted Vj . Ψ(k)

j,e is a weight that is calculated from the ratio ofthe inverse distances from the particle k to the vertices Ke of the element e inwhich it is located:

Ψ(k)j,e =

Πn 6=j |x(k)p,i − xn,i|∑

r∈KeΠm 6=r|x(k)

p,i − xm,i|(11)

The volumetric force excerted on the gaseous phase is then:

Fd,i =1Vj

∑k∈Dj

Ψ(k)j,e m

(k)p

(dup,idt

)(k)

(12)

The mass transfer rate per unit volume is calculated from:

Γ = − 1Vj

∑k∈Dj

Ψ(k)j,e

(dmp

d t

)(k)

(13)

and the heat transfer per unit volume is obtained by the following expression:

Π =1Vj

∑k∈Dj

Ψ(k)j,e

(dmp

d ths,F (Tp) + λπdpNu (Tp − T )

ln(BT + 1)BT

)(k)

(14)

2.2.2 Euler-Euler

In the Euler-Euler (EE) method, the spray is considered as a continuum, forwhich transport equations can be written. The conserved variables are formu-lated for the so-called mesoscopic quantities (noted with a ˘-operator), whichrepresent the conditional ensemble average over all droplets present in a givencontrol volume (Fevrier et al. [9]). In the formulation used here, the spray isassumed to be locally monodisperse. Furthermore, the effects of the randomuncorrelated motion are neglected. In the filtered equations used in a LES,Favre filtering is applied to an arbitrary mesoscopic quantity f :

αlfl = αlfl (15)

where αl is the spatially filtered liquid volume fraction. The equation for theconservation of the droplet number density nl then reads:

∂nl∂t

+∂nlul,j∂xj

= 0 (16)

The conservation equation for liquid mass becomes:

∂ρlαl∂t

+∂ρlαlul,j∂xj

= −Γ (17)

7

The term Γ describes the mass transfer per unit volume of the spray. It is theEulerian equivalent to equation 7:

Γ = ρlαl[π dSh [ρDF ] ln (1 +BM )

](18)

The equation for the liquid phase momentum is:

∂ρlαlul,i∂t

+∂ρlαlul,iul,j

∂xj= −Γul,i + Fd,i +

∂τ tl,ij∂xj

(19)

Fd,i =ρlαlτp

(ug,i − ul,i) (20)

where the droplet relaxation timescale τp is obtained by equation 6 and τ tl,ij isa subgrid stress tensor that is modeled using a Smagorinsky formulation [25]for the trace-free part together with a Yoshizawa formulation for the trace [17].The conservation equation for the liquid phase sensible enthalpy hs,l is:

∂ρlαlhs,l∂t

+∂ρlαlul,ihs,l

∂xi= −Π (21)

which is based on the same derivation as the evolution of droplet temperaturein the Lagrangian formulation (equation 9). In the present implementation,the subgrid-scale terms for liquid phase sensible enthalpy are assumed to benegligible. The term Pi on the right-hand side is the heat transfer per unitvolume between liquid and gaseous phase. In the Eulerian framework, it takesthe form:

Π = −Γhs,F (T l) + λπnldNu(T l − Tg

) ln(BT + 1)BT

(22)

In both EL and EE, the viscosity µ and the heat capacity at constant pressureCP depend on properties of the gaseous mixture surrounding the droplet. In thescope of the evaporation model, these properties that vary between the dropletsurface and the far-field are assumed to be constant and evaluated using the1/3-rule (Hubbard et al. [11]).

3 Numerical methods

3.1 Numerical methods for the gaseous phase

The gaseous LES solver AVBP explicitly solves the compressible Navier-Stokesequations with a centered scheme that achieves up to third-order accuracy inboth time and space [6]. In the present work, the second-order accurate Lax-Wendroff scheme [18] is used due to its lower numerical cost and sufficient accu-racy for the gas phase. To damp spurious oscillations, second-order and fourth-order artificial viscosities are applied [15]. A standard Smagorinsky formulation[25] is used to model subgrid-scale turbulence. Inlet and outlet boundary condi-tions are imposed using the NSCBC method [19]. Solid boundaries are modeled

8

using logarithmic wall laws [23, 12]. The gaseous AVBP solver can be combinedwith two different modules for the dispersed, liquid phase that shall be describedin the following.

3.2 Numerical methods for the dispersed, liquid phase

Euler-Lagrange

The interpolation of gaseous properties to the droplet is based on a linearleast-squares operator. For time advancement, the Lagrangian solver relies ona first-order forward-Euler method. Since the timestep is determined by theacoustic CFL for the gaseous solver, first order accuracy is deemed acceptablefor the temporal resolution of the liquid phase equations. To reduce computa-tional cost and memory requirements, droplets can be grouped into “parcels”.This can be justified for cases with large numbers of droplets. In the presentwork, a parcel contains 10 physical droplets.

Euler-Euler

In the EE framework, the multi-dimensional upwinding scheme PSI [27]scheme is used, as its robustness makes it very well-adapted for cases thatinclude spray injection. For unsteady problems, it is of first order in space andtime [1]. Despite its low accuracy, it has proven to yield good results in particle-laden homogeneous isotropic turbulence as well as industrial-scale applications[21, 15].

4 Liquid phase injection

The liquid injection system consists in a series of plain, round jets enteringthe injector perpendicularly to the airflow where they are atomized. This cor-responds to the problem of a liquid jet in a gaseous crossflow that includescomplex mechanisms of primary and secondary breakup [29, 4]. Because thebasic assumptions (dilute spray, no droplet breakup) do not hold in the near-field of the injection, the Eulerian and Lagrangian methods previously presentedbecome valid only after the resulting spray reaches a fully developed state in thefar-field. As it is not the main interest of the present application to capture thenear-field effects (i.e. primary and secondary breakup), the objective is to finda simple method, with little or no additional cost and to provide a sufficientlyaccurate result in the far-field. The relevant quality criteria are a correct spatialdistribution of the resulting spray plume (in terms of penetration height andlateral diffusion) as well as a realistic diameter distribution, both locally andspatially. If these criteria are satisfied, the spray that leaves the injector andenters the combustion chamber can be expected to behave realistically in termsof interaction with the turbulent air flow, evaporation and - in reactive cases -

9

interaction with the flame front.

Preliminary studies on the liquid jet in a crossflow case have been carriedout by Jaegle et al. [13, 14] using Eulerian and Lagrangian methods in thesame numerical framework as in the present study. Different momentum fluxratios q = ρlw

2l,inj/(ρgu

2g) between the liquid jet at injection and the crossflow

are investigated (ranging from q = 2 to q = 18). This study shows that asimple model for the liquid column region yields good results for the far-fieldspray distribution (compared to experimental data by Becker et al. [3]). Inthe Lagrangian framework, despite neglecting secondary breakup, good resultsfor the spatial diameter distribution in the far-field are obtained by injecting alog-normal size distribution that corresponds to the fully developed spray. Forthe locally monodisperse Euler-Euler method used here, injecting a spatiallyvarying diameter profile is of limited success only for a high momentum fluxratio case (q = 18).

In the present study, the momentum flux ratio has a value of q = 0.2. Effectsof the liquid column can therefore be neglected. Furthermore, in the Euleriancase, a constant diameter is injected as no improvement can be expected from avariable profile based on the findings of the preliminary study. In the Lagrangianformulation, two cases are chosen: in the first one, a monodisperse spray isinjected to create conditions that are identical to the Eulerian case for a directcomparison between EL and EE methods. In the second, a polydisperse spraywith a log-normal distribution is injected. The Sauter mean diameter of thisdistribution is based on a correlation provided by Becker et al. [3].

5 Simulation setup

5.1 Grid / boundary conditions

The computational grid is a critical contributor to the quality of the results inthe present study. Figure 3 shows an overview of the mesh. It is of unstructured,hybrid type, composed of tetraedral and prismatic and pyramidal elements. The(triangular) prisms form a single, closed layer in all regions where wall functionsare used to model the turbulent boundary layer. This layer facilitates the ap-plication of wall functions in the no-slip formulation but is is also advantageousin terms of the overall number of cells, because the near-wall grid refinementcan easily be controlled by adapting the prism aspect ratio without obtainingan excessive number of near-wall tetraedra.

The grid has several refined zones inside and downstream of the injector.The swirler channels are optimized for the use of wall-functions, with a targetfor near wall prismatic layer thickness of y+ ≈ 100. Areas of high grid resolutionare located in the pilot and main bowl, stretching outwards in areas where themain shear layers between the counter-rotating swirler stages or between thehigh-velocity regions and recirculation zones are located. Furthermore, the grid

10

Figure 3: Computational grid.

Mesh type unstructured, hybridCell types tetraedrae (domain volume)

prisms (domain boundary, areas with wall model)

Grid nodes 1 619 357Grid cells 8 540 311

Table 1: Mesh parameters

is refined in proximity of the cooling films.Grid refinement is significantly relaxed downstream of the high-shear regions

at the inejctor exit as well as inside the plenum. The only focus region related tothe two-phase simulations is a local refinement around the multipoint injection.The same mesh is used for all calculations presented in this study, both gaseousand two-phase, regardless of the liquid phase approach. Global parameters ofthis common mesh are summarized in table 1.

11

Case # 1 2 3 4Two-phase flow No EL EL EESpray injection No Polydisperse Monodisperse Monodisperse

Table 2: Summary of all simulation cases.

Gaseous phaseMass flux (plenum inlet) 0,377 kg/sTemperature 473 K

WallsWall temperature 473 K

Liquid phaseliquid mass flux (Multipoint inj.) 10 g/sliquid phase temperature 300 K

Table 3: Boundary values, liquid and gaseous phase

5.2 The cases considered

6 Results

6.1 Gaseous flow field

6.1.1 Qualitative description

The flow topology is explained at the in figure 4

6.1.2 Averaged quanities - comparison to the experiment

A validation against experimental data is needed to exclude errors in the gaseousflow that would interfere with the analysis of the spray. Averaged velocity pro-files are presented for the axial as well as the tangential component in figures 5and 6 respectively. These profiles are obtained along three measurement lineslocated at 10, 15 and 30 mm downstream of the swirler exit (see figure 4).The main peaks in axial and tangential velocity correspond to the main flowregion from the swirler and are well reproduced both in magnitude and posi-tion. Secondary peaks on the axial velocity profiles near the lateral walls arecreated by the cooling films. They are well captured on the first profile andslightly diffused at the 15 mm position. At 30 mm, both peaks have mergedand agreement is again excellent.

For the profiles of axial and tangential velocity fluctuations, shown in figures7 and 8 respectively, agreement is very good. On all measurement positions, theshape and magnitude of the fluctuations is well reproduced. For the tangential

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10! 15! 30 mm!

Shear layers!

Detached boundary

layer!

PVC!Central recirculation

zone!

Figure 4: Instantaneous field of axial velocity on the mid-plane. Position of themeasurement lines.

-60

-40

-20

0

20

40

60

y [

mm

]

6040200-20 uax [m/s]

-60

-40

-20

0

20

40

60

y [

mm

]

6040200-20 uax [m/s]

-60

-40

-20

0

20

40

60

y [

mm

]

6040200-20 uax [m/s]

10 mm 15 mm 30 mm

Figure 5: Mean axial velocity profiles of the gaseous flow. Comparison of LESresults (—), and experimental data (◦ ◦ ◦)

fluctuations, a slight over-estimation of the peak values can be observed.

Overall, results of the purely gaseous simulations are of very good quality.They match experimental data on first and second order statistics accurately.

13

-60

-40

-20

0

20

40

60

y [

mm

]

-40 -20 0 20 40 utan [m/s]

-60

-40

-20

0

20

40

60

y [

mm

]

-40 -20 0 20 40 utan [m/s]

-60

-40

-20

0

20

40

60

y [

mm

]

-40 -20 0 20 40 utan [m/s]

10 mm 15 mm 30 mm

Figure 6: Mean tangential velocity profiles of the gaseous flow. Comparison ofLES results (—), and experimental data (◦ ◦ ◦)

-60

-40

-20

0

20

40

60

y [

mm

]

403020100 uRMS,ax [m/s]

-60

-40

-20

0

20

40

60

y [

mm

]

403020100 uRMS,ax [m/s]

-60

-40

-20

0

20

40

60

y [

mm

]

403020100 uRMS,ax [m/s]

10 mm 15 mm 30 mm

Figure 7: Axial velocity fluctuation profiles of the gaseous flow. Comparison ofLES results and experimental data (◦ ◦ ◦)

This lays an excellent groundwork for the following study of two-phase flowbecause errors related to the prediction of the gaseous flow can be consideredto be negligible.

14

-60

-40

-20

0

20

40

60

y [

mm

]

2520151050 uRMS,tan [m/s]

-60

-40

-20

0

20

40

60

y [

mm

]

2520151050 uRMS,tan [m/s]

-60

-40

-20

0

20

40

60

y [

mm

]

2520151050 uRMS,tan [m/s]

10 mm 15 mm 30 mm

Figure 8: Tangential velocity fluctuation profiles of the gaseous flow. Compari-son of LES results and experimental data (◦ ◦ ◦)

15

6.2 Two-phase flow results

6.2.1 Qualitative description

Figure 9: Close up views of the multipoint injection. Left: isometric view of the(numerical) particle field in the chamber, polydisperse case. Right: EE results,iso-surface of the liquid volume flux to visualize the spray boundary.

Euler-Euler Polydisperse Euler-Lagrange

Figure 10: Comparison of EE results with polydisperse EL results. Left: iso-contours of liquid volume flux Φlv[m3/(sm2)] on the plane y = 0. Right: nu-merical particles in a 5 mm neighbourhood of the plane y = 0.

6.2.2 Averaged quanities - comparison to the experiment

6.3 Computational cost

The computational cost associated with EE and EL methods is an importantpart of the global comparison. Table 4 lists the key figures of the three simulation

16

Polydisperse Euler-Lagrange Monodisperse Euler-Lagrange

Figure 11: Comparison of polydisperse and monodisperse EL results. Left andright: particles in a 5 mm neighbourhood of the plane y = 0, shaded with theparticle diameter dp [m].

Euler-Euler Monodisperse Euler-Lagrange

Figure 12: Comparison of EE results with monodisperse EL results. Left: dropletdiameter field dl [m] on the plane y = 0, overlaid with the iso-contour markingthe clipping diameter. Right: numerical particles in a 5 mm neighbourhood ofthe plane y = 0, shaded with the droplet diameter dp [m].

cases. The key finding is a significantly higher cost of the EE simulation, wherethere is almost a factor 2 relative to the polydisperse EL case. Interestingly, thenumber of particles in the monodisperse EL case is slightly higher compared toits polydisperse counterpart, which leads to a slight advantage of the latter.

The apparent superiority of the EL approach in terms of CPU cost must,however, be relativized. One aspect is the use of the parcel approach for EL,which considerably accelerates the method for high particle numbers. The sec-ond is the longer time that is needed to achieve a given degree of statistical

17

40x10-3

20

0

-20

-40

z [

m]

-40x10-3 -20 0 20 40 y [m]

Figure 13: Placement of the sampling locations for histogram data in the y-zplane.

150x103

100

50

0

nor

m. p

roba

bilit

y

50x10-640302010 dp [m]

1.6x106

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

nor

m. p

roba

bilit

y

50x10-640302010 dp [m]

300x103

250

200

150

100

50

0

nor

m. p

roba

bilit

y

50x10-640302010 dp [m]

0.8

0.6

0.4

0.2

0.0

nor

m. p

roba

bilit

y

420400380360340 Tp [K]

0.10

0.08

0.06

0.04

0.02

0.00

nor

m. p

roba

bilit

y

420400380360340 Tp [K]

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0.20

0.15

0.10

0.05

0.00

nor

m. p

roba

bilit

y

420400380360340 Tp [K]

10mm 30mm 50mm

Figure 14: Diameter and Temperature histograms of the monodisperse EL sim-ulation. Samples taken in a series of cubes at 15mm downstream location and10mm (left), 30mm (middle) and 50mm (right) lateral distance from the cen-terline. (see figure 13 for the exact arrangement).

CPU timesEE EL monodisp. EL polydisp.

Number of numerical particles - 1 282 155 1 115 579Averaging time [s] 16.1 · 10−3 43.6 · 10−3 41.5 · 10−3

Av. time / residence time [−] 0.412 1.12 1.06CPU hours / res. time [−] 473 255 24764 processors on a SGI Altix ICE 8200 EX (Jade, CINES)

Table 4: Summary of the CPU times of two-phase-flow simulations of the TLCconfiguration

18

150x103

100

50

0

nor

m. p

roba

bilit

y

50x10-640302010 dp [m]

80x103

60

40

20

0

nor

m. p

roba

bilit

y

50x10-640302010 dp [m]

140x103

120

100

80

60

40

20

0

nor

m. p

roba

bilit

y

50x10-640302010 dp [m]

0.8

0.6

0.4

0.2

0.0

nor

m. p

roba

bilit

y

420400380360340 Tp [K]

50x10-3

40

30

20

10

0

nor

m. p

roba

bilit

y420400380360340

Tp [K]

0.3

0.2

0.1

0.0

nor

m. p

roba

bilit

y

420400380360340 Tp [K]

10mm 30mm 50mm

Figure 15: Diameter and Temperature histograms of the polydisperse EL simu-lation. Samples taken in a series of cubes at 15mm downstream location and10mm (left), 30mm (middle) and 50mm (right) lateral distance from the cen-terline. (see figure 13 for the exact arrangement).

Euler-Euler Monodisperse Euler-Lagrange

Figure 16: Comparison of EE results with monodisperse EL results. Left andright: field of the source term for mass transfer Γ [kg/(m3 s)] on the plane y = 0.

convergence, especially for RMS values. The comparison of averaging time overthe residence time in the chamber between EL and EE shows roughly a factor 2in favour of the EE method although the term “degree of convergence” is some-what arbitrary. Here, the use of the parcel approach would actually acceleratethe convergence of Lagrangian statistics. It depends on the computer resourcesand the sover scaling characteristics on parallel architectures if this preferrableis advisable or not.

19

-60

-40

-20

0

20

40

60

y [

mm

]

6040200-20 ul,ax [m/s]

-60

-40

-20

0

20

40

60

y [

mm

]

6040200-20 ul,ax [m/s]

10 mm 15 mm

Figure 17: TLC configuration - axial liquid phas velocity profiles. Comparisonof (inherently monodisperse) EE-results (—), monodisperse EL-results (- - -),polydisperse EL-results (····) and experimental data (◦ ◦ ◦).

-60

-40

-20

0

20

40

60

y [

mm

]

-40 -20 0 20 40 ul,tan [m/s]

-60

-40

-20

0

20

40

60

y [

mm

]

-40 -20 0 20 40 ul,tan [m/s]

10 mm 15 mm

Figure 18: TLC configuration - tangential liquid phas velocity profiles. Compar-ison of (inherently monodisperse) EE-results (—), monodisperse EL-results (- --), polydisperse EL-results (····) and experimental data (◦ ◦ ◦).

7 Conclusion

Acknowledgements

F. Jaegle gratefully acknowledges the support by the European communitythrough Marie Curie Fellowships (contract MEST-CT-2005-020426).

20

-60

-40

-20

0

20

40

60

y [

mm

]

403020100 ul,RMS,ax [m/s]

-60

-40

-20

0

20

40

60

y [

mm

]403020100

ul,RMS,ax [m/s]

10 mm 15 mm

Figure 19: TLC configuration - axial liquid phas velocity fluctuation profiles.Comparison of (inherently monodisperse) EE-results (—), monodisperse EL-results (- - -), polydisperse EL-results (····) and experimental data (◦ ◦ ◦).

-60

-40

-20

0

20

40

60

y [

mm

]

2520151050 ul,RMS,tan [m/s]

-60

-40

-20

0

20

40

60

y [

mm

]

2520151050 ul,RMS,tan [m/s]

10 mm 15 mm

Figure 20: TLC configuration - tangential liquid phas velocity fluctuation pro-files. Comparison of (inherently monodisperse) EE-results (—), monodisperseEL-results (- - -), polydisperse EL-results (····) and experimental data (◦ ◦ ◦).

21

-60

-40

-20

0

20

40

60

y [

mm

]

40x10-63020100 SMD [m]

-60

-40

-20

0

20

40

60

y [

mm

]

40x10-63020100 SMD [m]

10 mm 15 mm

Figure 21: TLC configuration - profiles of the Sauter mean diameter. Compar-ison of (inherently monodisperse) EE-results (—), monodisperse EL-results (- --), polydisperse EL-results (····) and experimental data (◦ ◦ ◦).

22

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