subgrid combustion modelling for large-eddy simulations

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2000 1: 209International Journal of Engine ResearchS Menon

Subgrid combustion modelling for large-eddy simulations

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Subgrid combustion modelling forlarge-eddy simulations

S MenonSchool of Aerospace Engineering, GeorgiaInstitute of Technology, Atlanta, Georgia, USA

Received 5 April 2000

Abstract: Next-generation gas turbine and internal com- liquid fuel atomization and increasing fuel–air

mixing are considered key issues to achieve efficientbustion engines are required to reduce pollutant emissionssignificantly and also to be fuel efficient. Accurate predic- combustion within a compact zone. However,

beyond proof of concept demonstration, optimizingtion of pollutant formation requires proper resolution ofthe spatio-temporal evolution of the unsteady mixing and combustor design will require understanding the

physics of the mixing process. However, the struc-combustion processes. Since conventional steady statemethods are not able to deal with these features, method- ture of the complex, three-dimensional, swirling

fuel–air mixing layers is very difficult to resolveology based on large-eddy simulations (LESs) is becominga viable choice to study unsteady reacting flows. This using current experimental and numerical methods.

From the numerical standpoint, since fuel atomiz-paper describes a new LES methodology developed recentlythat has demonstrated a capability to simulate reacting ation and fuel–air mixing are both highly unsteady,

conventional steady state methods cannot be used toturbulent flows accurately. A key feature of this newapproach is the manner in which small-scale turbulent elucidate the finer details. In addition, although

unsteady mixing processes can be studied quite accu-mixing and combustion processes are simulated. This fea-ture allows proper characterization of the effects of both rately using direct numerical simulation (DNS),

application of DNS is limited to flows at low to mod-large-scale convection and small-scale mixing on the scalarprocesses, thereby providing a more accurate prediction of erate Reynolds numbers (Re), typically of order 1000,

due to computational constraints (whereas Re in achemical reaction effects. LESs of high Reynolds numberpremixed flames in the flamelet regime and in the distrib- typical gas turbine combustor can be as high as

100 000). Thus, there is a need for a numericaluted reaction regime are used to describe the ability of thenew subgrid combustion model. approach capable of simulating unsteady turbulent

combustion in full-scale combustors with an accu-

racy that is acceptable from the engineering designKey words: combustion modelling, emissions, large-eddysimulations, engines, turbulent flows standpoint.

A simulation approach that has become quite

popular in recent years is large-eddy simulation

1. Introduction (LES). In LES, all scales larger than the grid reso-

lution are numerically simulated using a space- and

time-accurate scheme, while the effect of scalesDesirable features for the next-generation gas turbine

and internal combustion engines are combustion below the grid resolution is modelled using a subgrid

model. LES modelling for reacting flows requiresefficiency, compact combustor size, reduced emis-

sions and stable combustion in the lean limit. All two seemingly independent (but, as shown below,

interdependent) model developments. Closurethese (sometimes conflicting) requirements imply

that optimization of the engine for specific appli- requirements for momentum and scalar transport are

contradictory and this makes closure problematic. Incations will require a combined experimental and

numerical design strategy. Currently, improving LES modelling of the momentum transport, the effect

209Int J Engine Research Ω Vol 1 Ω no. 2JER 00300 Ω © 2000 Ω IMechE at MCGILL UNIVERSITY LIBRARY on November 22, 2013jer.sagepub.comDownloaded from



S Menon

of the unresolved small scales (assumed to be mostly where w is any physical variable and T is the time

period of the integration. In practical applications,isotropic) on the resolved motion is modelled using

T is finite, which implies that only time scales smalleran eddy viscosity-based subgrid model. This

than T have been filtered out. If all turbulent scalesapproach is acceptable for momentum transport

are smaller than T, then time-averaging will resultsince almost all the energy-containing scales are

in the mean flow field that has no turbulence-likeresolved, and the unresolved small scales primarily

variation. On the other hand, any slow variation (i.e.provide dissipation for the energy transferred from

at time scales larger than T if it exists) is allowed.the large scales. However, even here there are new

Applications of time-averaging to the governingconstraints to consider when attempting LES of high-

equations result in unknown terms that represent theRe flows. For example, if the small scales are to be

filtered scales of motion. Closure models for theseprimarily dissipative, the grid resolution must

terms have been developed and used for quite someresolve all scales up to the dissipation scales. For

time. However, even in non-reacting flows it hashigh-Re flows, this requirement can make compu-

been difficult to characterize the effects of all turbu-tational cost unacceptable (even on massively paral-

lent length and time scales by just one scale (as esti-lel systems) for engineering studies (as discussed

mated using the most popular k–e closure model).below).

In most applications, the model constants areFor non-premixed combustion to occur, fuel and

adjusted in order to obtain good agreement. Thisoxidizer species must mix at the molecular level. This

approach is questionable when new conceptualmixing process is dominated by turbulent mixing

designs are to be studied, since the appropriateand molecular diffusion in the small scales. Thus,

model constants are not known a priori.extending the eddy viscosity concept to scalar fields

The RANS approach also has a major difficulty inby using an eddy diffusivity model can be erroneousobtaining a closure for the reaction rate term.except under very specialized circumstances. ForProbability density function (PDF) methods, such aspremixed systems, accurate prediction of the flamethe assumed PDF or the more formal scalar-PDFstructure and its propagation characteristics is criti-method (which evolves the joint scalar PDF as a partcal. Since flame thickness can be very small, reso-of the solution rather than assuming it), have beenlution of the flame structure in a conventionalable to close the reaction rate term within the contextapproach is computationally expensive.of the RANS method [2]. A unique feature of theThis paper describes a new approach thatPDF method is that it can close the chemical pro-addresses some of these concerns. Section 2 brieflyduction/destruction term without requiring anysummarizes the formulation of the dynamic subgridmodelling. However, PDF methods cannot deal withclosure of momentum and energy transport and inmolecular diffusion effects and, in addition, in mostSection 3 a new subgrid scalar closure model forapplications (except for scalar velocity PDFhigh-Re reacting LES is described. Section 4 discussesmethods), the velocity field is determined using thethe application of the conventional and new LESstandard RANS closure for the momentum transport,approaches to high-Re turbulent reacting flows andwhich implies that the limitations in the fluid dynam-is followed by conclusions in Section 5.ics closure cannot be avoided.

A variant of the RANS method that is becoming

2. Closure Issues for Reacting LES popular is called unsteady RANS. In this paper, this

approach is distinguished from LES for a variety ofNumerical modelling of reacting flows can be reasons. By definition of time-averaging (or ensem-broadly categorized into three distinct approaches of ble-averaging), all scales smaller than T are modelledincreasing computational cost. The ‘cheapest’ (i.e. by the closure and the only unsteadiness allowed isrelative to the other two schemes) is the popular at time scales larger than T. Thus, RANS models thatmethod of Reynolds-averaged Navier–Stokes were originally developed for steady state appli-(RANS) that has been the backbone of current com- cations cannot be used for an unsteady solution,

putational fluid dynamics (CFD) applications in both especially if T is now considered smaller than some

non-reacting and reacting flows. In RANS, classical of the turbulent time scales.

time-averaging is defined as [1] Another unsteady RANS method that has been

proposed employs a ‘time-accurate’ RANS turbu-

lence closure. This approach requires that the turbu-w: (x

i)=

1

T P T

0

w(xi, t) dt (1)

lence model be developed using a methodology that

210 Int J Engine Research Ω Vol 1 Ω no. 2JER 00300 Ω © 2000 Ω IMechE at MCGILL UNIVERSITY LIBRARY on November 22, 2013jer.sagepub.comDownloaded from




takes into account the interaction between the unre- resolved strain rate tensor (as in the algebraic eddy

solved and resolved scales. An example of this viscosity models). However, in the latter case, equi-

approach is the k–e closure developed using renor- librium between subgrid turbulent kinetic energy

malization group (RNG) theory [3]. In the original production and dissipation (inherent in the algebraic

formulation, the constants of this model were deter- eddy viscosity model) cannot exist. As a result,

mined using the RNG technique. This method was subgrid closure requires a model capable of dealing

denoted as very large-eddy simulation (VLES) [3] to with the non-equilibrium effects in the small scale.

contrast it with unsteady RANS and the more con- This can be achieved by using a velocity scale that

ventional LES approach. adjusts to the local non-equilibrium effect. A model

At the other end of the spectrum is DNS, which based on the transport equation for the subgrid kin-

resolves all length (and time) scales from the largest etic energy is an obvious choice.

energy-containing scale (e.g. integral l ) to the small- Dynamic models, which predict the ‘coefficients’

est dissipative scale (Kolmogorov, g). However, even of the subgrid model as a part of the solution, have

for non-reacting flows, estimates show that the also been developed using Germano’s mathematical

number of grid points required to resolve all the identity [6]. This approach requires filtering at two

length scales goes as Re9/4, which means that even levels: the grid level and the test filter level (typically

for a moderate Reynolds number of Re=104 the grid twice the grid resolution). Scale similarity between

points needed for a DNS is N=109. This require- the properties at the two filter levels is required,

ment, along with the fact that to obtain data for stat- which implies that both the grid and test filter level

istical analysis sufficient time evolution of the flow cutoff must be in the inertial range (this requirement

field must be simulated, makes DNS of even non- also imposes a restriction on the grid resolution

reacting, moderate-Re flows impossible for the fore- allowed). Although this approach has improved pre-seeable future. Most DNS studies, as a result, are dictions in non-reacting flows, there remain someconfined to simple flows and to low Re (O(103)) limitations of this approach. The application offlows. Germano’s identity results in mathematical and

The extension of DNS to reacting flows is even numerical problems as described elsewhere [7, 8].more problematic. In flows of practical interest, the Many fixes have been suggested in the literature,flame structure can be very thin (in the premixed such as averaging in the homogeneous directions [9],flamelet regime the flame thickness, d

Lcan be orders evaluation along Lagrangian paths [10], etc.

of magnitude smaller than smallest turbulent scale, However, for complex high-Re flows in practicalg). Thus, even when the Kolmogorov scale is systems, there are no homogeneous directions andresolved, thin flames cannot be resolved. Two the fluid motion may be quite complex, makingapproaches have been attempted to circumvent this Lagrangian averaging difficult. These issues becomelimitation: the use of modified chemistry to artifici- even more relevant in reacting flows, since a turbu-ally thicken the flame in order to resolve it [4] and lent flame is a highly fluctuating three-dimensionalthe use of a thin flame model where the flame front surface and averaging across the flame isis tracked without resolving it [5]. Significant insight inappropriate.into flame–turbulence interactions has been ob- In this paper, a new dynamic closure that avoidstained using these approaches. However, high-Re the problems noted above is described and is com-flames as in real devices remain beyond the scope bined with a new subgrid combustion closure toof DNS. develop an LES capability to deal with high-Re

LES lies between the methodologies of RANS and reacting flows.DNS. In LES, characteristic length and velocity scales

have to be specified for the unresolved field. The

local grid size (D9 ) is an appropriate choice for the 3. Simulation Model for Momentum andcharacteristic length scale. The exact choice of the Energy Transportlength scale in LES determines how much of the kin-

The Navier–Stokes equations that govern the conser-etic energy is unresolved. If the cutoff is in the dissi-

vation of mass, momentum and energy in a fluid arepation range then negligible energy remains

filtered to obtain the LES equations for fluid motions.unresolved, but if the cutoff is in the inertial range

Spatial filtering reduces the high wave numberthen a significant portion of the kinetic energy is

Fourier components of the flow variables and separ-unresolved. In the former case, the characteristic vel-

ocity scale can be estimated using D9 and the local ates the resolved scale components from the




S Menon

unresolved scale components. Following reference dard heat of formation at temperature T0 and cp,m

is

the mth species specific heat at constant pressure.[11], the flow variables are decomposed into the

resolved (supergrid scale) and unresolved (subgrid In the LES equations, t:ij

and q:i

are approximated

simply in terms of the filtered velocity. The unclosedscale) components: f= f+ f ◊, where ˜ denotes

resolved and ◊ denotes subgrid quantities. The Favre subgrid terms representing, respectively, the subgrid

stress tensor, subgrid heat flux, unresolved viscousfiltered variable is then defined as f=r f /r: , where

the overbar represents spatial filtering, which is work and the species–temperature correlation are

defined as f:(xi, t)=∆ f(x∞

i, t)G

f(x

i, x∞

i) dx∞

i. Here, G

fis

the filter kernel and the integral extends over the

entire domain. Applying the filtering operation (in

the present study, a low-pass filter of the compu-

tsgsij

=r: (u0 iu

j− u

iu

j)

H sgsi

=r: (E0ui−Eu

i)+(pu

i−p:u

i)

ssgsi

=ujt

ij− u

jt:

ij

hsgs= ∑N

m=1

Rm

(T0 Ym−T Y0

m)

tational mesh is used and thus the characteristic size

of this filter is the grid width D9 ) to the Navier–Stokes

equations, the LES equations for mass, momentum

and energy are obtained.(6)

The LES closure of the scalar conservation equa-

tions is carried out in this study using a conventionalIn the present study, the unresolved viscous work

and a new subgrid mixing approach and is thereforessgs

iand the temperature–species correlation hsgs are

addressed in a separate section.neglected and only the subgrid stress and heat flux

are modelled. Earlier studies [12] showed that ssgsi

is3.1 LES equationsnegligible. The term hsgs can be closed only if theThe compressible LES equations are

temperature–scalar correlation in the subgrid scale is

known. The subgrid combustion model described inqr:qt

+qr: u

iqx

i

=0 (2)this paper has the capability to provide a closure

for hsgs.

qr: ui

qt+

qqx

j

(r: uiu

j+p:d

ij−t:

ij+tsgs

ij)=0 (3)

3.2 Subgrid closure for momentum and energy transport

In this study, the subgrid stresses and heat flux are

closed using a model based on the transport equationqr: Eqt

+qqx

i

[(r: E+p:)ui+q:

i− u

jt:

ji+H sgs

i+ssgs

i]=0

for the subgrid kinetic energy [13–17]:

(4)

qr:ksgs

qt+

qqx

i

(r: uiksgs)=P sgs−D sgs+

qqx

iAr:n

tPr

t

qksgs

qxiBIn the above equation, r is the mass density, p is the

pressure, E is the total energy per unit mass, uiis the (7)

velocity vector, qi

is the heat flux vector and dij

is

the Kronecker delta. The viscous stress tensor is where ksgs=12(u0 2

k− u2

k) is the subgrid kinetic energy

given by tij=m(qu

i/qx

j+qu

j/qx

i)−2

3m(qu

k/qx

k)d

ij, and Pr

tis the turbulent Prandtl number. The terms

where m is the molecular viscosity coefficient and is on the right-hand side of equation (7) represent,determined using Sutherland’s law. The pressure is respectively, the production, the dissipation and thedetermined from the equation of state for a perfect transport of the subgrid kinetic energy. The pro-gas duction term is modelled as Psgs=−tsgs

ij(qu

i/qx

j),

where the subgrid shear stresses tsgsij

are evaluated as

p:=r: T ∑N

m=1

Rm

Y0m+hsgs (5)

tsgsij

=−2r:nt(S

ij−1

3S

kkd

ij)+2

3r:ksgsd

ij(8)

Earlier studies split the subgrid stress into threewhere Rm

and Ym

are respectively the mth species

gas constant and mass fraction. The total energy per components: the Leonard stress, the cross term and

the Reynolds stress, with each term being indepen-unit volume is determined from rE=r(e+12u2

k),

where e is the internal energy per unit mass given dently modelled. However, it was shown [18] that

this approach is not Galilean invariant and that theby e=WNm=1

Ym

hm−p/r. Here, h

mis the mth species

enthalpy. The caloric equation of state is given by modelled LES equations no longer represent the

filtered Navier–Stokes equations. As a result, mosthm=Dh0

f,m+∆T

T0

cp,m

(T ∞) dT ∞, where Dh0f,m

is the stan-





modern closures model tsgsij

directly in order to all these small-scale processes. Such a model is

described in this paper.maintain Galilean invariance. The same approach is

followed here.4.1 LES scalar modelIn the above equation, n

tis the subgrid eddy

The scalar conservation equations can also be filteredviscosity given by nt=C

n(ksgs)1/2D9 and S

ij=

(as done for the momentum and energy equations)12(qu

i/qx

j+qu

j/qx

i) is the resolved-scale rate-of-

to obtain the LES resolved equations. These equa-strain tensor. The dissipation term is modelled astions are discussed below, although in the presentD sgs=C

er: (ksgs)3/2/D9 . The two coefficients appearing

study a new approach is employed to circumventin the above equations, Cn

and Ce, can be determined

some of the problems encountered while attemptingusing a localized dynamic approach. Earlier,closure of this equation. On applying the LES filter,Germano’s dynamic approach was used to deter-the LES-resolved equations can be derived asmine these coefficients [19, 20]. However, this

approach also encounters the same numerical prob-

lems seen in the dynamic algebraic closure. qr:Y0m

qt+

qqx

i

[r:Y0m

(ui+V

i,m)]+

qY sgsi,m

qxi

=vm

(9)An alternative dynamic approach was recently

developed and is used in the present study. DetailsHere, V

i,mand v

mare, respectively, the mth speciesof this localized dynamic k-equation model (denoted

diffusion velocity (given by Fick’s law) and the prod-hereafter as LDKM) is described elsewhereuction/destruction term. Also, Ysgs

i,m=r: (Y0m

ui−Y0

mu

i)[14, 15, 21] and therefore is avoided here for brevity.

is the subgrid scalar flux term. Both vm

and Y sgsi,mIt has been shown earlier that the LDKM avoids the

require modelling.problems encountered in the earlier dynamic formu-

lation and allows fully localized (i.e. without requir-4.2 LES flamelet model

ing any special averaging technique) evaluationThe scalar LES equations described above can be

without encountering any numerical instability.used for both non-premixed and premixed combus-

Analysis of the results [15] have shown that thetion. However, under special considerations, simpli-

LDKM is Galilean-invariant and satisfies the realiz-fications can be used to reduce the computational

ability conditions [22]. From a computational stand-effort. In particular, for premixed combustion a com-

point, the cost of the present dynamic procedure isputationally very efficient model can be used in the

not significant (about the same as the dynamic modelflamelet regime which is often encountered in practi-

of Germano et al. [6]) due to its simplicity. Thecal combustion devices. In this regime, the flame

additional computational cost is primarily due to thethickness (d

L) is much smaller than g and the charac-

inclusion of a transport equation for ksgs. For non-teristic burning time (t

c) is much smaller than the

reacting flows, using reasonable grid resolutions thecharacteristic flow time (t

t). As a result, the flame

effects of the subgrid model on the statistical quanti-structure remains laminar and the flame is a thin

ties are usually marginal. However, it becomes sig-front propagating at a speed dictated by the mixture

nificant when coarse grids are used as demonstratedproperties, which is wrinkled and convected by the

in recent studies [21, 23]. The capability of the LDKMflow.

approach in coarse grid LES was also demonstratedA model equation that describes the propagation

in an independent study [24].of a thin flame by convective transport and normal

burning (self-propagation by Huygens’ principle) is

the G equation [25]:

4. Simulation Model for Scalar TransportqrGqt

+qru

iG

qxi

=−rSL|VG | (10)

Scalar fields are affected by three physical processes:

advection (due to the velocity field), molecular dif-

fusion and chemical reactions. While the filtered vel- where G(x, t) is a progress variable that defines the

location of the flame and SL

is the local unstretchedocity field causes large-scale convection of the scalar

fields, the subgrid velocity fluctuations lead to fine- laminar flame speed. In the flow field, G is prescribed

in the range [0, 1] with a value of unity in thescale mixing. Furthermore, molecular diffusion and

reaction processes are also small-scale phenomena unburned region and zero in the burnt region, with

the flame identified by a fixed value of 0<G0<1.and hence are affected strongly by the fine-scale

mixing. The scalar subgrid model must account for Therefore, equation (10) describes the kinematic bal-




S Menon

ance between the convection of a level surface, where ScG is a turbulent Schmidt number (which can

also be computed using a dynamic procedure [29]).defined as G=G0, by the fluid velocity and normal

propagation at a speed SL. In this flame model, the Note that the gradient closure approximation seems

to violate the physics of turbulent transport givenflame structure is effectively ignored since only

the propagating surface is modelled. As a result, the that counter-gradient diffusion can dominate the

transport of scalar fluxes. However, counter-gradientdetails on the reaction rates and species diffusion can

be ignored. transport is a large-scale phenomenon [30] and in the

LES methodology, the large scales are directly com-Using mass conservation [26], rSL

can be replaced

by r0S0

L, where r

0is the reference reactant density puted. Therefore, counter-gradient diffusion should

be accounted for despite the subgrid closureand S0L

is the undisturbed laminar flame speed. Note

that S0L

is constant while SL

increases through the assumed.

For premixed combustion in the flamelet regime,flame [27]. Upon filtering, equation (10) gives

a subgrid flame speed model that is a function of u∞and S

Lhas proven quite successful [21] in closing theqr:G

qt+

qqx

i

(r: uiG)=−Ssgs−

qGsgsi

qxi

(11)source term Ssgs: Ssgs=r

0S0

L|VG |#r

0u

t|VG |, where

ut

is the turbulent flame speed. This speed is not

known explicitly and, therefore, must be modelled.Here, the unclosed subgrid closure terms rep-

Two different turbulent flame speed models haveresenting, respectively, the filtered source term and

been evaluated in recent studies. The first model isthe unresolved transport term are summarized as

the RNG model [31], ut/S0

L=exp[(u∞

sgs)2/(u

t)2],follows:

which relates ut

to the laminar flame speed and the

subgrid turbulence intensity u∞sgs

=u∞=EIksgs. ThisSsgs=r0S0

L|VG |

G sgsi

=r: (u0iG− u

iG)

model compares well with experimental data in the

low to moderately high u∞/SL

range [27]. In addition,(12)

the model predicts the rapid increase in ut/S

L(at low

u∞/SL) and then a gradual bending (at higher u∞/S

L).

The LES G equation (11) must be solved along withAnother model correlated data over a wide range of

the LES equations since G is coupled with the u∞/SL

[32]:thermodynamic variables through the filtered

internal energy e=cvT+Dh

fG. Here Dh

f= u

tS

L

=A1+bu∞2S2

LB1/2

(14)cp(T

p−T

f) is the heat of formation and T

pand T

fare

the product and fuel temperatures respectively. From

this approximation, it can be seen that temperature where b is an adjustable parameter. In earlier studiesvaries linearly with G, i.e. T=(e−Dh

fG)/c

v. This [21, 29], these two models were evaluated using

means that the temperature profile is approximately experimental data [33]. It was observed that bothas wide as the G profile. Earlier studies [28] have models perform well in the low to moderate u∞/S

Lshown that although the linear dependence of T on limit. Also, Pocheau’s model with b=20 was foundG results in distributed heat release, since flame to fit the experimental data in the high u∞/S

Llimit as

propagation speed is explicitly included, it does not well. Comparisons of both these models werecause significant error. reported recently [29].

As shown below, the new subgrid combustion A final point to note is that the flame speed closuremodel has the ability to capture ‘thin’ flames without described above requires the knowledge of theencountering the aforementioned numerical spread- subgrid turbulence intensity u∞. This field is naturallying of the flame on the LES grid. available when the ksgs subgrid model (LDKM) is

employed. In contrast, when the algebraic eddy vis-4.3 Conventional closure for scalar transport

cosity model is used there is no way to determine u∞.Regardless of the approach (i.e. the species or the

This is another advantage of the ksgs model describedG equation), closure of the scalar–velocity corre-

in this paper.lation (Y sgs

i,mor G sgs

i) can be modelled using a gradient

assumption. For example, G sgsi

can be modelled as 4.4 Subgrid scalar or flame simulation model

The above closure is a conventional approach in

which the scalar equations are filtered and theG sgs

i=r: (u0

iG−u:

iG)#−

r:nt

ScGVG (13)

unclosed terms are modelled. However, this





approach fails to explictly account for the small-scale The subgrid method for solving equations (16) and

(17) or (18) involves (a) conducting LEM model simu-processes that directly impact scalar fields. In this

section, a new subgrid scalar simulation approach is lations in each of the three-dimensional LES grid

cells, (b) transporting the subgrid scalar fields acrossdescribed that addresses the fundamental limitations

of the conventional approach. the LES cell faces to account for large-scale advection

and (c) coupling the subgrid heat release effect to theIn this approach, a subgrid simulation of the scalar

field is carried out within every LES cell. The scalar momentum and energy transport via volumetric

expansion. These processes are summarized below.fields within the subgrid field evolves due to small-

scale processes of molecular diffusion (or flame4.4.1 Subgrid processes

propagation for the flamelet model), turbulent stir-Flame propagation, volumetric expansion due to

ring and volumetric expansion due to heat release.heat release and subgrid turbulent stirring due to the

The subgrid simulation model is a variant of thesubgrid velocity field are modelled using one-dimen-

linear-eddy model (LEM) developed earlier [34–36].sional simulations within each LES cell. The

This method is called LES–LEM hereafter.reduction of the subgrid domain to one dimension

Consider the following generic form of an unfil-is critical in reducing the overall computational cost.

tered scalar (temperature, species concentrations)However, as shown in the cited literature, the LEM

evolution equation:model has the capability to capture the realistic three-

dimensional turbulence effect on the scalar fields inqCqt

=−u0k

qCqx

k

−u∞k

qCqx

k

+qqx

jCD

c

qCqx

jD+w

c(15) spite of the one-dimensional representation that is

used to characterize the scalar fields in the subgrid.

The characteristic time scales associated with theseHere, u0

kand u∞

kare the resolved and unresolved vel-

subgrid processes can be very small, and time stepsocity fields, D

cis the diffusion coefficient and w

cis

much smaller than DtLES

(used for integration of thea representative source term. In the LES–LEM

momentum equation in LES) may be required forapproach, a two-scale numerical procedure is used

subgrid modelling. In fact, all subgrid and supergridwhich can be formally represented as

processes evolve concurrently at their respective

time scales. Therefore, one large-scale advection stepC*−Cn

DtLES

=−u0k

qCqx

k

−(u∞k)face

qCqx

k

(16) [equation (16)] may be followed by many smaller

time steps to integrate the subgrid processes.

The reaction–diffusion equations on the one-Cn+1−C*dimensional domain for the mth chemical species can

be written as= P t+DtLES

tCu∞

k

qCqx

k

+qqx

jAD

c

qCqx

jB+C

sD dt (17)

qrYm

qt=

qqs ArD

m

qYm

qs B+vm+(F

m)stir

(19)Equation (16) represents the large-scale advection of

the scalar field by the resolved velocity field and is

modelled by transferring fluid volumes between the and the equivalent flame propagation model using

control volumes on the three-dimensional grid. Also, the G equation is

(u∞k)face in equation (16) represents the component of

the subgrid velocity field (on the control volume qGqt

=−SL|VG |+(F

G)stir

(20)faces) that causes volume (scalar) transport between

LES cells.Here, s denotes the subgrid one-dimensionalIn the case of the G equation, the first step in thedomain. For multi-species reacting flows with heatintegration scheme [equation (16)] remains the samerelease, a one-dimensional temperature equation of(G replaces C) but the reaction and diffusion termsthe formin the second step [the last two terms on the right-

hand side of equation (17)] are replaced by the

propagation term, −SL|VG |. The resulting subgrid qT

qt=−

1

rcp

∑N

m=1

cp,m

Ym

Vm

qTqs

+1

rc:p

qqs Ak:

qTqsBequation is a forced propagation equation for G:

−1

rc:p

∑N

m=1

hmv

mW

m+(F

T)stir

(21)Gn+1−G*= P t+DtLES

tCu∞

k

qGqx

k

−SL|VG |D dt (18)




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is also solved, along with the species equation.

Since the subgrid velocity field is assumed to be

isotropic, its effect on the scalar fields is symbolically

denoted by (Fw

)stir

and is implemented by a stochastic

Monte Carlo simulation within the one-dimensional

domain. The ‘subgrid’ velocity field intensity u∞ is

needed for implementation of this stochastic stirring

process. Again, ksgs naturally provides this

information.

The reaction–diffusion scalar equations or the one-

dimensional G equation is solved on the one-dimen-

sional domain using the standard finite difference

approach. The orientation of the one-dimensional

domain is in the direction of the scalar gradient since

it can be shown [36] that an accurate estimate Fig. 1 Triplet mapping procedure used to stir the scalar field.

of scalar dissipation [x=2D (qC/qxk)2] can be A segment l chosen, divided into three equal parts and

obtained as 2D (qC/qs)2 using LEM. This orien- then the spatial gradients are increased by a factor of 3

tation is used to determine the length of the subgrid and the middle segment inverted. This rearrangement

domain Dl, as described elsewhere [37]. The number leaves the mean value and the scalar values at the ends

of subgrid LEM cells (NLEM

) used to discretize Dl

is unchanged, but the scalar gradient is increased three-determined to ensure that even the smallest subgrid fold. As shown below, if the LEM line is in the flamescale (e.g. g) is fully resolved [38]. Thus, the reso- normal direction, the triplet map increases the numberlution is fine enough to allow a subgrid ‘DNS’. This of crossings of a single scalar value and thereby mimicsimplies that no explicit closure of the production the effect of a turbulent eddy on the scalar field, whichterm v

mis required. For the G-equation approach, may be interpreted as an increase in the flame area

subgrid flame propagation at the local laminar flame caused by wrinkling.speed S

Lcan be modelled without requiring any clos-

ure as well.field. Figure 1 gives a schematic representation of theThe explicit resolution of the production term v

mtriplet map. The details of this mapping procedurein LES–LEM is similar to the capability in scalar–PDFhave been discussed elsewhere [34, 36], where it wasmethods [2]. However, LES–LEM also includes ashown that this mapping can capture correctly thecapability to deal with molecular diffusion (includ-physical increase in scalar gradient (without affectinging differential diffusion) which is not available inthe mean scalar concentration) due to eddy motion.PDF models. This capability allows the present

Implementation of turbulent stirring via tripletsubgrid model to capture Lewis and/or Schmidtmapping requires three parameters: (a) the size ofnumber dependencies without requiring any ad hoceddy that is causing the stirring, (b) the location ofmodifications to the model.this eddy within the one-dimensional domain andThe exothermic nature of the flame leads to volu-(c) the frequency of the stirring process. The size ofmetric dilatation which causes the flow to accelerate.

the eddy is obtained from an eddy size distributionThe effect of dilatation (on the scalar field) is mod-

(PDF), f(l ), and its location is randomly chosen (fromelled within the one-dimensional domain. The

a uniform distribution) within the one-dimensionalthermodynamic pressure is assumed to be constant

domain. Finally, the event frequency l is determinedlocally within the one-dimensional domain and

from inertial range scaling laws as applicable totherefore density is inversely proportional to the

three-dimensional turbulence. These parameterstemperature. When the density within an LEM cell

have been described in detail elsewhere [34, 36, 38]changes due to burning, mass is conserved by

and therefore are not discussed further.adjusting (increasing) the volume (and length) of

In summary, the subgrid simulation model concur-that LEM cell and regridding the subgrid domain.

rently involves modelling the flame propagation viaAdditional details are given elsewhere [37, 39, 40].

equation (20) or reaction diffusion via equation (19)Turbulent stirring [denoted by (Fw

)stir

] is

and turbulent stirring. Stirring interrupts scalar pro-implemented using a series of stochastic rearrange-

cesses at instants determined by l (stirring time inter-ment events called ‘triplet maps’ [34], each of which

represents the effect of a single eddy on the scalar val is Dtstir

=1/lDl) and is carried out by choosing





an eddy size l from f(l ), randomly locating this eddy evolution are modelled either through the advection

scheme or by subgrid modelling, the integration ofwithin the one-dimensional domain (using a uniform

distribution) and then applying the triplet mapping the scalar equations on the three-dimensional LES

grid is not required. This is in contrast to the conven-process.

The unique feature of the turbulent stirring used tional LES approach where the filtered scalar conser-

vation equations are modelled using a numericalhere is worth emphasizing. Although the subgrid

LEM domain is one-dimensional, the subgrid stirring discretization scheme similar to that used for the

momentum equations.parameters are determined using inertial range sca-

ling laws for fully three-dimensional turbulence.

Thus, the effect of turbulent stirring on the scalar5. Results and Discussionfield mimics physically realistic mixing in three-

dimensional flows. This capability allows LEM toIn this section, some of the results obtained using

capture many scalar properties accurately.this new subgrid combustion formulation are dis-

cussed. First the ability of the subgrid combustion4.4.2 Large-scale advectionmodel is demonstrated in order to capture quantitat-The advection of the subgrid scalar fields by the LES-ively the premixed flame structure in flamelet andresolved velocity field is modelled by solving equa-distributed reaction regimes. This demonstration istion (16) using a Lagrangian algorithm [38, 40, 41]important since, within typical devices such as gasthat is quite similar to the volume of fluid (VOF)turbine combustors and internal combustion engines,method [42]. This method involves the transfer ofthe nature of combustion can change dependingfluid (i.e. LEM cells) between LES control volumesupon local conditions. Thus, non-premixed, partiallyto account for the volume flux across the cell faces.premixed and premixed combustion can all occurUsing the velocities at the LES cell faces, thesimultaneously. In premixed combustion, dependingvolume (scalar) flux crossing each cell face in timeupon the local chemical and turbulent time scales,Dt

LESis computed. The direction of transfer is deter-

combustion can occur in the flamelet and in the dis-mined by the velocity on the cell faces and thetributed reaction regime (now called the thin reactionnumber of LEM cells are chosen to account for thezones regime). Subgrid models must be able to dealvolume to be transferred. There are some specificwith changes in the flame and combustion character-numerical issues regarding the implementation ofistics without any ad hoc changes.the volume transfers and the manner by which the

Subsequently, two problems are addressed: a stag-hyperbolic (front tracking) nature of the scheme isnation point premixed flame and a highly swirlingachieved. Details are given in the cited literature.flame in a full-scale gas turbine engine. The stag-Both passive advection of a sharp interface or out-nation point flame is used to demonstrate the abilityward normal propagation (as in a propagating pre-of the new subgrid combustion model by comparingmixed flame) can be captured quite accurately bywith the predictions using a conventional closure,this method.and the combustion in the full-scale combustor is

used to demonstrate the ability of LES to predict4.4.3 Subgrid–supergrid coupling

The subgrid small-scale and the large-scale advection ‘engineering’ flows of practical interest.

It is worth noting that the computational cost ofprocesses detailed in the above two sections account

for the evolution of the scalar field. To integrate the reacting flow LES is considerable, even without

using the LES–LEM approach and, as a result, suchscalar fields with the resolved-scale momentum

dynamics, volumetric dilatation is required. The simulations require parallel processing. The

LES–LEM approach is highly optimal for parallelLES-resolved scalar variables (Y0k

and T) are obtained

by ensemble-averaging the scalar field in the local implementation and has been optimized and

implemented on parallel systems using MPI libraryone-dimensional subgrid domains. For example, the

filtered temperature T is obtained as T=WrT/Wr. directives. The code achieves nearly linear scale-up

with an increase in the number of processors [21]. AHere, W denotes a summation over all the subgrid

LEM cells within a given LES cell. The subgrid tem- typical conventional LES (with the G equation) using

500 000 points requires 2 GB of memory and on aperature field T is obtained either by solving the tem-

perature equation (21) or by using the relation 128-processor CRAY T3E requires around 10 000

single processor hours to obtain sufficient flow-between G and T.

Since all the physical mechanisms that effect scalar through times for statistical analysis. The LES–LEM




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approach using the same grid and with 100 LEM Recent studies [45–47] have suggested that even

in the distributed reaction regime the reaction zonecells per LES cell will increase the cost by a factor of 3.

is very thin and of the order of the laminar flameAlthough this computational time estimate

thickness. This result is derived from the obser-appears prohibitive, it is worth noting that the rapid

vations that, for large Karlovitz number Ka=(df/g)2,increase in CPU (central processing unit) speeds

Reynolds number Re=u∞L/n and Damkohler(especially in the Intel CPUs) in recent years suggests

number Da=tt/t

c, turbulent eddies can enter thethat future simulations could be carried out effec-

preheat zone and thus increase turbulent transporttively on PC-based clusters at a fraction of the current

of heat and species away from it. This redistributioncost. It is estimated that, using the next-generation

can thicken the preheat zone. However, eddies doPC cluster with a high-speed cluster switch, LES of

not penetrate into the reaction zone since they aretypical systems of engineering interest could be car-

dissipated by increased viscous dissipation near theried out within a matter of days. However, to achieve

flame. Using these results, the distributed reactionsuch a rapid turn-around the grid resolution must

regime has been reclassified as the extended flameletbe relatively coarse, which will require subgrid clos-

[45] or the thin reaction zones [48] regime. The keyure that is more physics based and capable of

implication is that modified versions of flameletincorporating the effect of missing scales accurately.

models can be used in the thin reaction zones regime,

5.1 Flame structure in flamelet and distributed reaction as recently described by Peters [48].

regimes Here, the ability of the LEM model to capture the

In premixed systems, in addition to the Reynolds flame structure in both the flamelet and distributed

number, other non-dimensional parameters become regime without any ad hoc fixes is addressed. In par-

relevant. The structure and propagation character- ticular, methane–air flames B1, F1, F2 and F3 are

istics of a turbulent premixed flame can be rep- simulated; these were recently experimentally stud-

resented in a combustion diagram such as the one ied [46, 47, 49]. The typical locations of these flames

shown in Fig. 2 [43, 44]. Two regimes are of particu- are given in Fig. 2. As shown, flame B1 is in the corru-

lar interest here. The flamelet regime is characterized gated flamelet regime while the flames F3 and F1

by df%g and thus turbulence can only wrinkle the cover the entire distributed regime. Only flames B1

flame without affecting its structure. On the other and F1 are discussed here.

hand, in the distributed reaction regime, g<df, tur- In order to obtain a realistic chemical state over

bulent eddies can penetrate into the flame, thereby a wide range of operating conditions, a 15-step,

modifying the flame structure. Flames in both these 19-reaction skeletal mechanism is employed (J.-Y.

regimes can exist in gas turbine and internal combus- Chen, personal communication, 1999). This mechan-

ism (which included NOx

kinetics) has been showntion engines.

Fig. 2 Diagram of turbulent premixed combustion regimes. The locations of the flames F1 and B1 discussed in this paper are shown.





to be quite accurate over a wide range of equivalence a part of the simulation and when the same composi-

tion reoccurs the table is searched and the storedratios. It is also capable of predicting extinction and

re-ignition, which is particularly relevant here since information is retrieved using the fast binary tree

search algorithm. Since only the accessible region isthe F1 flame is considered close to the extinction

limit [47]. stored, the overall time required to build, retrieve

and store information reduces significantly. AThe numerical method is the same as in the earlier

study [27, 50]. Since the chemical reaction point prob- speed-up of over 30 is obtained in the chemistry cal-

culation using the ISAT procedure. Further details oflem is very stiff and computationally very expensive

in the present study, in situ adaptive tabulation the ISAT algorithm are given elsewhere [51] and are

therefore avoided here for brevity.(ISAT) is used efficiently to handle finite rate kinetics

[51]. In ISAT, only the accessed region of the com- Typical instantaneous images of the flame struc-

ture in terms of the temperature and the destructionposition space, which is a subset of the whole realiz-

able region, is tabulated. This tabulation is done as rate of methane, vCH

4

, for flames B1 and F1 are

Fig. 3 Instantaneous temperature and vCH

4

for the B1 and F1 flames. Note that the scale of vCH

4

is changed for ease of presentation.




S Menon

shown in Figs 3a and b respectively. For comparison, zones, with a thick preheat zone in front, is character-

istic of flames in the thin reaction zone regime.a typical experimental snapshot of the temperature

and CH in the F1 flame is shown in Fig. 4. The rate Mean and variance of the reaction progress vari-

able, c=(T−Tu)/(T

b−T

u), where T

uand T

bare,v

CH4

is used to identify the location of the reaction

zone (CH is not available in the skeletal mechanism) respectively, the reactant and product temperatures,

have also been computed. The transition from flame-since it has been shown to correlate well with it [52].

The transition from the flamelet (flame B1) to the let to thin reaction zone regime combustion is best

presented by the probability density function (PDF)structure seen in the thin reaction zones regime is

clearly apparent in these figures. For flame B1, the of c. Figures 5a and b present the PDFs at six different

locations for flames B1 and F1 respectively. Aspreheat zone upstream of the flame zone is inert,

with its temperature close to the free stream value. expected, in the flamelet regime (flame B1), the PDF

exhibits two peaks corresponding to unburnt andHowever, as Ka and Re increase, eddies penetrate

into the preheat zone and increase the temperature burnt gases respectively. In this regime, probability

of the reactive states corresponding to the transitionahead of the flame brush. For flame F1, the tempera-

ture in the preheat zone increases. However, the reac- between reactants and products is small.

For the F1 flame, the PDF is not bimodal since thetion zone still remains localized and thin in all cases,

with vCH

4

peaking at the location of steep tempera- interaction between the flame and turbulence is not

entirely kinematic but also depends on the lengthture gradients. There is very good qualitative agree-

ment between the temperature profiles and the scales. Small eddies that enter the preheat zone

broaden the flame, increasing the probability of inter-experimental data [47]. The reaction zone thickness

is estimated to be 0.16, 0.21 and 0.27 mm for flames mediate values of the progress variable. There is

excellent qualitative agreement with the progressF1, F2 and F3 respectively. These values are very

close to the laminar flame thickness of 0.175 mm [46]. variable PDFs reported in reference [53] for flames

in similar regimes.Experimental data [47] for CH suggests a value in

the range 0.2–0.5 mm for the F3 and F1 flames.

Although the reaction zone is thin and of the order 5.2 LES of stagnation point flames

To demonstrate the ability of LES–LEM in premixedof the laminar flame thickness, the preheat zone is

much larger. Using the definition [47] that the pre- cases, stagnation point flames were simulated. The

stagnation point flame has been used extensively inheat zone is between the temperatures 600 and

1300 K, the thickness of the preheat zone is estimated the laboratory to study stationary turbulent pre-

mixed flames and consists of premixed reactantsas 2.2, 1.92 and 1.92 mm for flames F1, F2 and F3

respectively. In the experiments [47] the preheat zone impinging on a flat plate. Upon ignition, the pre-

mixed flame is formed above the plate and becomesthickness was estimated to be in the range 2–3.5 mm

for flame F1. The presence of such thin reaction stationary at a location where its upward propa-

Fig. 4 Instantaneous temperature and CH emission for the flame F1 as measured in the experiments [47].





Fig. 5 PDF of the progress variable c at six different locations in the flames. The transition from flamelet structure (bi-model PDF)

for flame B1 to the thin reaction zones flame structure in F1 is apparent in these PDFs.

gation into the reactant due to burning is balanced 89×129×129 was used for the conventional case

and a grid of 69×89×89 (with 100 LEM cells in eachby the incoming turbulent flow. The flame structure

and propagation characteristics can be adjusted by LES cell) was used for the LES–LEM case. The

coarser grid for LES–LEM is acceptable since thechanging the fuel–air mixture and/or the flow speed.

Focus here will be on burning in the wrinkled flame- subgrid domain is well resolved.

Since the flame is in the corrugated flameletlet regime which the internal flame structure is

undisturbed and the reaction zone is a single con- regime, it should be a relatively thin front that is

wrinkled by the incoming turbulence. Figure 6nected domain. For these simulations a grid of




S Menon

prediction of turbulence is much better in the case

of LEM–LES. Across the flame, the density decreases

significantly, which causes the flow to accelerate

tremendously across the flame. This results in very

high (flame normal) velocities on the product side

(compared to the reactants side). The unsteady oscil-

lations of the flame at any given point in the flame

brush thus causes very high intermittency, which in

turn leads to an increase in u∞.This physics is captured quite accurately by the

LEM–LES method. In the conventional LES (Fig. 6a),

the flame structure has a finite thickness which is

determined by the numerics (grid, scheme, etc.). As

a consequence, the flow acceleration (due to the den-

sity drop) is much more gradual than in the case of

LES–LEM. This smoothing of the flow gradients

reduces the flow intermittency and hence the con-

ventional approach does not produce the peak in u∞at the flame brush as in the experiments and the

LES–LEM simulation. This problem is also typical of

LES, which uses finite rate chemistry and transport

equations for chemical species on the LES grid. Note

that the resolution of any real flame on a three-

dimensional grid is impossible using current compu-

tational resources. Thus, the flamelet-type burning

may never be captured using conventional methods.

On the other hand, the LEM–LES method appears to

achieve this goal due to a combination of the features

of the subgrid LEM and the front tracking scheme.Fig. 6 Comparison of conventional LES and LES–LEM predic-

tions in the stagnation point flow. The premixed reac-5.3 LES of full-scale combustor flowtants are moving to the right and the flat wall is at theFigure 8 shows the schematic of a General Electricright edge of the figure. As shown, LES–LEM capturesDLE LM 6000 combustor. In the experimental versionthe flame as a thin wrinkled flame whereas the conven-of this combustor, high-swirling, lean premixed reac-tional LES numerically diffuses the flame.tants (methane–air) enter the square combustor (the

shape was primarily dictated by experimentalcompares the conventional LES prediction with

measurement requirements) from a circular pipe atLES–LEM. As shown, LEM–LES captures the flame

a preheated condition of 600 K. The test conditionsas a thin wrinkled front whereas the flame is cap-

are typical of actual full-scale operation: Re (basedtured over a broader zone by the conventional LES

on inlet diameter) of 330 000, swirl number of 0.56,(using the RNG-based Yakhot model described in

Karlovitz number of 42 and Damkohler number of 8.Section 4.3). The finite difference scheme in the con-

A relatively coarse grid of the order of 97×65×81ventional LES leads to smoother wrinkles whereas

was employed along with the LDKM closure. Inflowthe stochastic nature of the (flame) area creation

conditions were prescribed using normalized pro-mechanism (subgrid stirring) in LEM–LES leads to a

files (provided by GE) for the mean velocity field.more wrinkled structure.

Inflow turbulence using a prescribed energy spec-A more quantitative comparison of the superior

trum is also added to account for turbulent fluctu-ability of LES–LEM can be obtained by comparing

ations. Further details are given elsewhere [21].the predictions with experimental data. The mean

Due to high swirl, the local turbulence is quite highand the r.m.s. velocity profiles predicted by the

and this test condition lies in the thin reaction zonesLEM–LES and the conventional LES are compared

regime. Therefore, a flame-broadened dynamicto experimental data [54–56] in Fig. 7. As shown, the

mean is predicted fairly well by both models but the subgrid model for LES was developed recently that





Fig. 7 Comparison of the predicted mean and r.m.s. axial velocity profiles with experimental data [54–56]. The conventional LES

was carried out using an RNG-based flame speed model [31].

combines the G equation with the Pocheau model for intensity) and the coordinates are non-dimensional-

ized, respectively, by the maximum mean axial vel-the turbulent flame speed ut

(a comparison of the

various models is discussed in reference [29]). Here, ocity at the inlet (U0) and the inlet jet diameter (D

0).

The dynamic Pocheau model (with b=20), whenthis model is used to simulate a test case for which

data were available for comparison. combined with the broadened-flame model (denoted

PBD20 in the figures), predicts the mean axial velocityFigure 9a shows the mean axial velocity variation

along the centre-line and Fig. 9b shows the mean variation quite accurately. Different values of b do

not significantly affect this agreement and b=20radial velocity at x/D0=0.18. Note that, hereafter,

all velocity components (both mean and turbulence shows the best agreement with the experimental




S Menon

Fig. 8 Schematic of the LM 6000 being tested at General Electric.

data. The agreement with the radial velocity is also Acknowledgementsreasonable.

This work was supported by the Army ResearchA stringent test of LES is the ability to predict the

Office under the Multidisciplinary Universityturbulent fluctuation levels. This ability is highly

Research Initiative. Computational time was pro-dependent on the accurate characterization of the

vided by DOD High Performance Computinginflow turbulence (not much is known about the

Centers at NAVO, MS and SMDC, AL under a DODinflow turbulence in the experiments). Figures 10a

Grand Challenge Project.and b show, respectively, the root mean square

(r.m.s.) profiles of the fluctuating axial (urms

) and

radial (wrms

) velocity components, generated in this Referencesregion due to the unsteady motion of the flame. The

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scalar transport. Physics of Fluids A, 1991, 11, 2746–2754.engines should become feasible in the near future.





Fig. 9 Prediction of mean axial and radial velocity variations in the LM 6000. Here, PBD20 denotes the test case that employed a

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