Flow, Turbulence and Combustion
72:
1–28, 2004.© 2004
Kluwer Academic Publishers. Printed in the Netherlands.
1
Large Eddy Simulation of Premixed TurbulentCombustion Using
Flame Surface WrinklingModel
G. TABOR
1
and H.G. WELLER
21
School of Engineering, Computer Science and Mathematics, Harrison Building, University of Exeter, North Park Road, Exeter EX4 4QF, U.K.; Email: g.r.tabor@ex.ac.uk
2
Nabla Ltd., The Mews, Picketts Lodge, Picketts Lane, Salfords, Surrey RH1 5RG, U.K.
Received 24 April 2002; accepted in revised form 5 May 2003
Abstract.
One commonlyused method for deriving the RANS equations for multicomponent ﬂowis thetechnique of conditional averaging. Inthis paper the concept is extended toLES,by introducingtheoperations of conditional ﬁlteringandsurface ﬁltering.Propertiesof theﬁlteredindicator function
b
are investigated mathematically and computationally. These techniques are then used to deriveconditionally ﬁltered versions of the Navier–Stokes equations which are appropriate for simulatingmulticomponent ﬂow in LES. Transport equations for the favreaveraged indicator function
b
and theunresolved interface properties (the wrinkling and the surface area per unit volume) are also derived.Since the paper is directed towards modelling premixed combustion in the ﬂamelet regime, closureof the equations is achieved by introducing physical models based on the picture of the ﬂame as awrinkled surface separating burnt and unburnt components of the ﬂuid. This leads to a set of modelsfor premixed turbulent combustion of varying complexity. The results of applying one of this set of models to propagation of a spherical ﬂame in isotropic homogeneous turbulence are analysed.
JEL Codes:
D24, L60, 047.
Key words:
Large Eddy Simulation, premixed turbulent combustion.
1. Introduction
PremixedTurbulent Combustion isahighly complex process, but onewhich greatlyaffects everyday life. The quest to understand the physical processes better is continual, and one aspect of it is the search for computational models to describethe processes involved. Such models must of necessity be less detailed than thephysical processes occurring in the system, but should aim to capture the essenceof these processes. In turn, the models can provide a greater understanding of theprocesses involved, and provide us with the ability to predict the behaviour of speciﬁc combustion systems. Thus they are of great importance in the design of combustion devices such as Internal Combustion (IC) engines and gas turbines.A working model of turbulent combustion must provide adequate treatments forthe turbulence, the chemical reactions of the combustion (and consequential heat
2
G. TABOR AND H.G. WELLER
release), as wellas the mutual interaction ofthese areas, since the combustion altersthe physical properties of the ﬂuid and drives the ﬂow, whilst the ﬂow moves reactants and products around and thus inﬂuences the combustion. About the simplestpossible model combines a Reynolds Averaged Navier–Stokes (RANS)descriptionof the turbulence with a simplistic model of the combustion which provides amodel for the heat release as a straightforward function of the reactant speciesconcentration (for example, the Eddy Breakup model of Spalding [31]). Numerousimprovements on these simple models have been investigated over the years, inparticular concentrating on improved methods for characterising the species concentration at apoint (and thus the prediction ofthe heat release) by PDFtechniques,or improved ﬂame modelling.Large Eddy Simulation (LES) of premixed turbulent combustion is an activearea of research. It offers the possibility of signiﬁcant improvements over RANS,interms ofaccuracy ofthe solution, the ability tohandle countergradient diffusion,and the provision of greater information about the turbulent ﬂow ﬁeld which renders irrelevant some of the modelling assumptions necessary in RANS combustionmodels. The approaches used in LESare based on various ways of computationallyfollowing the ﬂame front. In premixed combustion the ﬂow consists of regions of unburnt reactants and regions of combusted products. The extent of combustionof the gas can be described in various ways in terms of a progress variable takingvalues between 0 and 1, with the extreme values indicating the presence of unburntor fully burnt phases, and the transition between these values marking the ﬂamefront. This can be linked directly to physical properties of the gas, for instance byutilising normalised temperature (
T
) or product mass fraction (
Y
):
c
=
T
−
T
u
T
c
−
T
u
or
c
=
Y
P
Y
P,b
.
(1)The exact linkage is not important however, and the progress variable can be considered simply as indexing the ammount of combustion, however deﬁned. In thispaper we will use a progress (technically a regress) variable
b
=
1
−
c
with
b
∈ [
0
,
1
]
, where 0 represents fully burnt gas and 1 unburnt gas.There is a problem here though. In the ﬂamesheet regime of high Damköhlerand Reynolds numbers the reaction zone is very thin indeed and the transition inthe progress variable is too sharp to be explicitly resolved on the LES mesh. Oneapproach sometimes employed is the Thickened Flame (TF) approach [7, 10]. Inthe TF method the ﬂame front is artiﬁcially thickened by multiplying the thermaland molecular diffusivities by a factor
F
and reducing the reaction rate by the samefactor. The result is a thickened ﬂame front with the same laminar ﬂame speed
S
l
, which can be resolved on the LES computational mesh, and thus its motioncan be calculated without additional SGS modelling. This has several advantages,simplifying the chemical reaction modelling and eliminating the need for ad hocsubmodels for ignition and ﬂamewall interactions. However it does involve altering the physics of the ﬂame front in a substantial manner. In particular the response
LES OF PREMIXED TURBULENT COMBUSTION
3of the ﬂame to unsteady phenomena and to strain induced by the velocity ﬁeld ismodiﬁed by the thickening procedure [1, 10].The mostcommon approach for LES,known as the
G
equation method, is basedon a levelset approach [3, 16, 19, 21, 22]. Here a function
G
is constructed to havethe property that the zero value isosurface represents the combustion interface.
G
is not related to the progress variable, so other values of
G
have no physical signiﬁcance and are merely chosen for computational convenience. A straightforwardtransport equation is then solved for
G
:
∂
G
∂t
+
U
.
∇
G
=
S
T
∇
G

,
(2)where
U
is the ﬂuid velocity and
S
T
the turbulent ﬂame speed, i.e. the rate of propagation of the ﬂame front due to combustion. The challenge in this approachcomes from developing adequate modelling for the turbulent ﬂame speed which isa welldeﬁned quantity that depends on local ﬂow conditions [24]. There are alsonumerical problems with the accurate propagation of
G
.The other option for simulation of combustion in this regime is to link theprogression of the ﬂame front to additional physical properties, e.g. geometricproperties of the surface. In the
G
equation, surface stretch and curvature effectsare treated by consideration of higher moments of
G
. An alternative class of models can be constructed based on solving for variables describing these geometricalparameters [33]. In RANS,the basic ‘laminar ﬂamelet’ models have been extended[8, 9, 20, 27]: the ﬂame front propagates locally as a laminar ﬂame but at thesame time is being wrinkled due to interactions with the turbulence. The ﬂamepropagation speed can be modelled in terms of the laminar ﬂame speed (a knownquantity) and the degree of wrinkling of the ﬂame at the point, given by the ﬂamearea per unit volume
. The system as a whole is described in terms of transportequations for the ﬁltered progress variable and for
. This approach has also beeninvestigated for LES [2, 15]. An alternative RANS model proposed by Weller[34, 35], represents the geometric properties of the ﬂame front in terms of thedensity of wrinkling
, which is the ﬂame area per unit area resolved in the meandirection of propagation. This choice of variable makes the modelling somewhateasier compared with the equivalent equation for
, for instance by separating outa term representing ﬂame annihilation by cusp formation. It also provides for aspectral analysis of the ﬂameturbulence interaction [36]. This RANS model wasformulated using the technique of Conditional Averaging [11]. The aim of ourcurrent work is to formulate an LES version of this model. In order to do so wemust introduce an analogous techique, that of Conditional Filtering, to derive thetransport equations for a multicomponent system. This technique is the subjectof this paper. Section 2 introduces the concept of Conditional Filtering in LES,and discusses the regularity of the ﬂame surface in relation to the surface ﬁltering process introduced as part of the analysis. The effect of ﬁltering a simulatedindicator function appropriate for combustion is investigated in 2d. In Section 3,Conditional Filtered versions of the Navier–Stokes Equations (NSE) are presented,
4
G. TABOR AND H.G. WELLER
together with transport equations for properties of the surface, in particular areaper unit volume
and wrinkling
. Finally, in Section 4 possible closures of theequations are discussed. The evolution of a spherical ﬂame is calculated using sucha model and its properties discussed.
2. Conditional and Surface Filtering
The Navier–Stokes Equations (NSE) for a compressible ﬂuid are
∂ρ∂t
+∇
.ρ
U
=
0
,∂ρ
U
∂t
+∇
.(ρ
U
⊗
U
)
=−∇
p
+∇
.
S
,∂ρe∂t
+∇
.ρe
U
= −
p
∇
.
U
+
S
.
D
+∇
.κ
∇
e,
(3)where
S
=
λ
∇
.
UI
+
2
µ
D
,
D
=
12
(
∇
U
+∇
U
T
).
(4)In Conditional Averaging in RANS, an indicator function is introduced [11] whichtakes the value 1 in the unburnt region phase and 0 in the burnt region. The NSEare multiplied by this function and then ensembleaveraged: the indicator functionprojects out one of the components, and so the resulting equation is for that component alone. This process introduces additional terms (in addition to the standardReynolds Stress term arising from the ensemble averaging process) which canbe written in terms of a surface average operation which represents the effect of the interface on the dynamics of the phase under consideration. These terms willcommonly require modelling. Transport equations can also be formulated in thisway for the ensemble averaged indicator function, which has the interpretation of the probability of ﬁnding the phase at that point, and for quantities relating to thesmallscale geometry of the interface.InLESitisassumed that thedependent variables intheNSEcan bedecomposedinto GS and SGS components, i.e.
ψ
=
ψ
+
ψ
. The GS component is obtainedby ﬁltering
ψ
, which is a convolution between it and a ﬁlter function
G
with theproperties
D
G(
x
)
d
3
x
=
1, lim
→
0
G(
x
,)
=
δ(
x
)
and
G(
x
,)
∈
C
n
(
R
3
)
with compact support. The decomposition into mean and ﬂuctuating componentsis thus analogous to the decomposition in RANS, but with differing interpretationsof the resulting variables. We can adapt LES to include the concept of conditionalaveraging by introducing an indicator function
I
which is a generalised function(or distribution) such that
I
(
x
,t)
=
1 if
(
x
,t)
is in phase A (say the unburnt gas),0 otherwise.(5)