Large Eddy Simulation of Premixed Turbulent Combustion Using E Flame Surface Wrinkling Model

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. TABOR1 and H.G. WELLER2

1School of Engineering, Computer Science and Mathematics, Harrison Building, University ofExeter, North Park Road, Exeter EX4 4QF, U.K.; E-mail: [email protected] 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 commonly-used method for deriving the RANS equations for multicomponent flowis the technique of conditional averaging. In this paper the concept is extended to LES, by introducingthe operations of conditional filtering and surface filtering. Properties of the filtered indicator functionb 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 favre-averaged 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 ofmodels 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

Premixed Turbulent Combustion is a highly complex process, but one which greatlyaffects everyday life. The quest to understand the physical processes better is con-tinual, 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 ofspecific combustion systems. Thus they are of great importance in the design ofcombustion 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 well as the mutual interaction of these areas, since the combustion altersthe physical properties of the fluid and drives the flow, whilst the flow moves reac-tants 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 con-centration at a point (and thus the prediction of the heat release) by PDF techniques,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,in terms of accuracy of the solution, the ability to handle counter-gradient diffusion,and the provision of greater information about the turbulent flow field which ren-ders irrelevant some of the modelling assumptions necessary in RANS combustionmodels. The approaches used in LES are based on various ways of computationallyfollowing the flame front. In premixed combustion the flow consists of regions ofunburnt 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 − Tu

Tc − Tu

or c = YP

YP,b

. (1)

The exact linkage is not important however, and the progress variable can be con-sidered simply as indexing the ammount of combustion, however defined. In thispaper we will use a progress (technically a regress) variable b = 1 − c withb ∈ [0, 1], where 0 represents fully burnt gas and 1 unburnt gas.

There is a problem here though. In the flame-sheet 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 speedSl, 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 flame-wall interactions. However it does involve alter-ing the physics of the flame front in a substantial manner. In particular the response

LES OF PREMIXED TURBULENT COMBUSTION 3

of the flame to unsteady phenomena and to strain induced by the velocity field ismodified by the thickening procedure [1, 10].

The most common approach for LES, known as the G-equation method, is basedon a level-set approach [3, 16, 19, 21, 22]. Here a function G is constructed to havethe property that the zero value isosurface represents the combustion interface. Gis not related to the progress variable, so other values of G have no physical sig-nificance and are merely chosen for computational convenience. A straightforwardtransport equation is then solved for G:

∂G

∂t+ U.∇G = ST |∇G|, (2)

where U is the fluid velocity and ST the turbulent flame speed, i.e. the rate ofpropagation of the flame front due to combustion. The challenge in this approachcomes from developing adequate modelling for the turbulent flame speed which isa well-defined 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 mod-els 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 flame-turbulence 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 filter-ing process introduced as part of the analysis. The effect of filtering a simulatedindicator function appropriate for combustion is investigated in 2-d. In Section 3,Conditional Filtered versions of the Navier–Stokes Equations (NSE) are presented,


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+ ∇.ρU = 0,

∂ρU∂t

+ ∇.(ρU ⊗ U) = −∇p + ∇.S,

∂ρe

∂t+ ∇.ρeU = −p∇.U + S.D + ∇.κ∇e, (3)

where

S = λ∇.UI + 2µD, D = 1

2(∇U + ∇UT ). (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 ensemble-averaged: the indicator functionprojects out one of the components, and so the resulting equation is for that com-ponent 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 ofthe 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 ofthe probability of finding the phase at that point, and for quantities relating to thesmall-scale geometry of the interface.

In LES it is assumed that the dependent variables in the NSE can be decomposedinto 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) d3x = 1, lim�→0 G(x,�) = δ(x) and G(x,�) ∈ Cn(R3)

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)


Figure 1. Local curvilinear coordinates for the flame surface. Here x⊥ is the direction normalto the flame surface, and (η, ς) coordinatize the flame surface.

Since I is a generalised function we can take its gradient. This will be zero every-where but on the interface, where it will have the direction of the outward facingnormal. If we introduce a coordinate system at the interface (x⊥, x‖) where x⊥ isthe direction normal to the interface and x‖ = (η, ς) are coordinates in the interfacemanifold (see Figure 1), then

I(x⊥, η, ς) = �(x⊥),

∇I = 1

h⊥∂⊥�(x⊥)n⊥ = 1

h⊥δ(x⊥)n⊥, (6)

where � is the Heaviside step function. Here h2⊥ is the appropriate element of

the metric tensor for this coordinate system, and thus relates to the curvature (andhence the wrinkling) of the surface. n⊥ is a unit vector in the x⊥ direction, i.e.normal to the actual surface.

We can define an interface velocity UI such that

dI

dt= ∂I

∂t+ UI .∇I = 0. (7)

Note that UI is defined here to be the total velocity of the interface irrespective ofwhether it is due to advection by the flow or generation (or removal) of phase A.From this we have the relation

∂I

∂t= − 1

h⊥UI .n⊥δ(x⊥). (8)


We can now introduce a ‘conditional filtering’ taking the form

ψ = G ∗ (Iψ) =∫D

G(x − x′)I(x′, t)ψ(x′, t) d3x′, (9)

for a tensor ψ of any rank. ψ is the phase-weighted value of ψ at any point. Thecombustion progress variable b can be introduced here as a GS indicator function

by writing ψ = b ψu , where b (x, t) is the probability of the point (x, t) being inthe unburnt gas (phase u). Setting ψ = 1, we find

b (x, t) =∫D

G(x − x′)I(x′, t) d3x′. (10)

Since combustion involves compressible flow, we need to deal with density varia-tions. Typically terms will involve the product ρψ , which we can write

ρψ = b ρψu (11)

and can define a density-weighted average ψu based on the u-phase to split up thissecond term:

ρψu = ρu ψu, (12)

thus giving

ρψ = b ρu ψu. (13)

We also need to be able to deal with spatial and temporal derivatives. If weassume a constant filter size (of course, issues may arise for non-uniform filters,see [12]),

∇ ◦ ψ = ∇ ◦ {G ∗ (Iψ)} = G ∗ {∇ ◦ (Iψ)}, (14)

where ∇◦ indicates an appropriate tensor derivative. Also

∂ψ

∂t= ∂

∂t{G ∗ (Iψ)} = G ∗ ∂(Iψ)

∂t. (15)

From filtering the identity

I∇ ◦ ψ = ∇ ◦ (ψI) − ψ ◦ ∇I, (16)

using Equation (6), we have

∇ ◦ ψ = ∇ ◦ ψ −∫D

G(x − x′)ψ(x′) ◦ n⊥ δ((x′ − xI ).n⊥)1

h⊥d3x′

= ∇ ◦ ψ − ︷︸︸︷

ψ ◦ n⊥� (17)


Here we have defined a surface filtering operation︷︸︸︷

as

︷︸︸︷

ψ = 1

�

∫D

G(x − x′)ψ(x′)δ((x′ − xI ).n⊥)1

h⊥d3x′ (18)

Because of the term δ((x′ − xI ).n⊥) this integral is non-zero only in the immediatevicinity of the interface. If we transform coordinates in this integral to the (x⊥, x‖)set, then we can perform the x⊥ integral, leading to

︷︸︸︷

ψ = 1

�

∫ ∫ ∫G⊥(x⊥ − x′

⊥)ψ(x′⊥, x′

‖)δ(x′⊥ − x⊥,I ) dx′

⊥

× G‖(x‖ − x′‖)|J| d2x′

‖

= 1

�G⊥(x⊥ − x⊥,I )

∫ ∫G‖(x‖ − x′

‖)ψ(x⊥,I , x′‖)|J| d2x′

‖,

assuming that the filter G can be split into components perpendicular and parallel tothe interface. |J| is the Jacobian for the (η, ς) coordinate system. In other words,the variable ψ is filtered only on the interface manifold, but the influence of theinterface is extended by the filter to its immediate neighbourhood. Writing ψ = 1in this relation and changing variables as before, gives

� = G⊥(x⊥ − x⊥,I )

∫ ∫G‖(x‖ − x′

‖)|J| d2x′‖. (19)

Since |J| d2x′‖ is the area element on the surface, � has the interpretation of beingthe ammount of interface within the filter support. In a similar way, the identity

I∂ψ

∂t= ∂ψI

∂t− ψ

∂I

∂t(20)

generates the commutation relation

∂

∂tψ − ∂ψ

∂t= ︷︸︸︷

ψUI .n⊥�. (21)

Finally, substituting ψ = 1 into Equation (17) gives us

∇b = ︷︸︸︷

n⊥�, (22)

relating the progress variable b (GS indicator function) to the surface filtereddirection vector

︷︸︸︷

n⊥. The identity

∂b

∂t= −︷︸︸︷

UI .n⊥� (23)

can be derived in a similar manner from Equation (21), or by filtering Equation (8).We can decompose the motion of the interface UI into the motion due to advection


and a term −van⊥ which is due to the advance of the interface relative to the flow,and so is due to the generation or destruction of phase A: UI = U + van⊥. Thisproduces the result

∂b

∂t+ ︷︸︸︷

U.n⊥� = −︷︸︸︷

va�. (24)

The cross term can be split as︷︸︸︷

U.n⊥ = ︷︸︸︷

U .︷︸︸︷

n⊥ +︷︸︸︷

v .n ⊥, where the hashes represent

surface fluctuating components. Writing Dc =︷︸︸︷

v .n

⊥� gives

∂b

∂t+ ︷︸︸︷

U .∇b + Dc = −︷︸︸︷

va� (25)

2.1. REGULARITY OF THE SURFACE

In the preceeding section it is implicitly assumed that the surface defined by theindicator function is well behaved. In this context, the surface is well behaved ifit is a 2-d submanifold of R3. Over the majority of the surface this will be thecase; however potentially we will need to consider separately a number of points atwhich problems arise. Pope [27] distinguishes four cases where the surface regular-ity breaks down: singularities, internal edges, self-intersections and critical points.Of these, internal edges do not occur in our case, since our surface delineates aboundary between distinct areas of space and is at all times closed.

Consider the case of two spherical fronts expanding towards each other un-til they touch, which is the intersection problem described by Pope. If I is Cn-continuous for the time being, then this corresponds to the formation of a saddlepoint in I . At such a point, ∇I = 0, and the surface is no longer parameter-isable. If the surfaces meet tangentially, the curvature at this point is infinite. Ifthe spheres merge further, it is conceivable that the point will become a line (ac-tually a ring) at which one of the radii of curvature is infinite. This could bedescribed as a negative line cusp. Other forms of propagation can also lead tothe development of other singularities at points or lines. However since it is area-based properties which are relevant for the front propagation, and these structureshave zero area, this should not affect the modelling. Consider the tip of a cone,which can be represented as part of a sphere of radius r: it has curvature 1/r2, butan area contribution r2 sin θ dθ dφ. Hence the contribution of this surface to anycurvature-related properties → 0 as r → 0.

Related problems arise with the calculation of b , and related smoothed vari-ables. The spatial smoothing removes completely the possibility of infinite surfacecurvature, but increases the likelyhood of partial or total overlap. Such overlap willcause problems if the propagation of the surface is linked to |∇b |, as it is in theWeller combustion models [34]. In this case the indicator function of the combinedflame will be different from those of the individual flames, leading to an incorrect


Figure 2. Filtering of flame front in 2-d. (a) gives the physical flame front as generated by‘wrinkles’, whilst (b)shows the Delaunay triangulation of this curve. (c) and (d) are the resultof filtering the indicator function using a Gaussian with σ = 5 and σ = 25 (the box dimensionis 100 units). Meshes of these cell sizes are also shown.

value for |∇b |, and an error in the propagation. This situation occurs for examplein the early stages of propagation of a spherical flame front, i.e. as the ignitionkernel propagates outwards. Thus at the start of the simulation appropriate ignitionmodelling must be provided. In the case presented here a very simplistic model isused in which the ignition energy is distributed over a small number of cells at thecentre of the computational domain, but this can probably be improved upon.


2.2. 2-D SIMULATION OF b

Weller et al. [36] present a model of a combusting flame front using a length-scaledecomposition of turbulence into eddies. The interaction of these individual eddieswith a closed curve can be calculated and the net effect of all eddies generated bysuperposition, resulting in a 2-d closed curve with similar geometric properties toa real turbulent flame: essentially a simulated snapshot of a slice through the flame.Weller et al. wrote a code (‘wrinkles’) based on this to generate and analyse flamefronts in 2-d. The original aim was to validate the spectral modelling used in theWeller RANS model. Here we use the output from this code is to look at the effectof spatial filtering on a simulated indicator function.

The wrinkles code outputs the flame surface (a line in 2-d) as a series of linesegments. The result of applying it to a circle is shown in Figure 2a, which is agood approximation to a spherical flame front in 2-d. The first step is to determinethe inside and outside of the closed curve, which is less easy than it sounds. Thepoints defining the curve were used as the input to a code ‘Triangle’, which is a2-d mesh generator using constrained quality conforming Delaunay triangulation[29, 30]. This generates a series of points (and associated triangulation) on theinside of the curve. The value 1 is then allocated to all these points, thus creatinga representation of the indicator function. To filter the indicator function, the areawas then sampled using a regular 2562 grid using the visualisation code VTK [28].The data on this regular mesh was then convolved with a 2-d Gaussian functionof width σ : exp (−r2/2σ 2) and the resulting function contoured. Figures 2c and2d show the result for σ = 5 and σ = 25 respectively. Meshes with cells ofsize σ = 5 and σ = 25 are superimposed for comparison. These show the gridscale appropriate for a real CFD calculation (see Section 4.2 below). The samplingplane was of side 100 units, so for the finest mesh each cell contains > 100 points,providing adequate resolution of the filtered indicator function.

Several points are demonstrated here. The filtered indicator function is resolvedby 2–3 mesh cells. This is slightly sharper than for actual calculations (see Section4.2) but not much. Meanwhile there is clearly a large degree of wrinkling which isnot resolved and which will be modelled by the wrinkling variable � derived in thenext section. Figure 2c would be appropriate for a well-developed, large sphericalflame, and clearly the overall structure is well resolved. Figure 2d could be taken asan earlier version of the flame shortly after ignition. With only 6 CFD cells acrossthe structure, b is inadequately resolved, and the two sides of the flame will interactin an undesirable manner. This demonstrates that the ignition and very early growthof the flame will be difficult to simulate, but once the flame has grown sufficientlythe model is adequate for the purpose. Although this is not a perfect analog of thereal situation, not least because the simulated surface is not propagating, it doesindicate the basic idea is sound.


3. Conditionally Filtered Equations

3.1. CONTINUITY EQUATION

We can now filter the NSE’s (Equations (3a–3c)) using these relations. Startingwith the continuity equation, we have

∂ρ

∂t+ ∇.ρU = �

︷︸︸︷

ρ(U − UI ).n⊥. (26)

Since UI = U + van⊥, and using Equation (12), Equation (26) can be written

∂b ρu

∂t+ ∇.b ρu Uu = −︷︸︸︷

ρva�. (27)

The definition of b provides a measure of the large-scale geometry of the sur-face. Substituting n⊥ into Equation (18) and using the same coordinate transfor-mation:

︷︸︸︷

n⊥ = 1

�G⊥(x⊥ − x⊥,I )

∫ ∫G‖(x‖ − x′

‖)n⊥(x⊥,I , x′‖)|J| d2x′

‖. (28)

Now n⊥|J| d2x′‖ is a directed area element on the surface, so︷︸︸︷

n⊥ has the interpreta-tion of being the ammount of directed interface within the filter support. This canbe related to the GS interface direction nf

︷︸︸︷

n⊥ = nf

�(29)

with a wrinkling factor � defined as

� = 1

|︷︸︸︷

n⊥| = �∣∣G⊥(x⊥ − x⊥,I )∫∫

G‖(x‖ − x′‖)n⊥(x⊥,I , x′

‖)|J| d2x′‖∣∣ . (30)

From this we see that � gives the total subgrid surface area divided by the smoothedsurface area, i.e. the wrinkling. nf is of course a unit vector in the same direction.Substituting Equation (22) gives

� = �

|∇b | (31)

with |∇b | the area of the grid scale surface.

3.2. MOMENTUM, ENERGY EQUATIONS

Similarly, conditional filtering the momentum equation gives

∂

∂t(b ρu Uu) + ∇.(b ρu Uu ⊗ Uu)

= −∇b pu + ∇.{b

(Su − Bu

)}+

[︷︸︸︷

(pI − S).n⊥ − ︷︸︸︷

ρvaU]�, (32)


Bu = (ρU ⊗ U)u − ρu Uu ⊗ Uu. (33)

The terms in [ ] in (32) represent the effects of the interface on the momentumbalance.

This leaves the energy equation. The l.h.s. of (3c) can be treated straightfor-wardly:

∂ρe

∂t+ ∇.ρeU = ∂b ρu eu

∂t+ ∇.b ρu euUu + ∇.

[b (ρeU)u − b ρu euUu

]+ ︷︸︸︷

ρeva�.

Following normal practice, we write b (ρeU)u − b ρu euUu = b b. The term on thethird line represents the contribution of the work done by the interface during itsmotion to the energy transport. The r.h.s. is more involved. We write

−p∇.U + S.D + ∇.h = −b (p∇.U)u + b (S.D)u + ∇.b hu

= − (b ρu ∇.Uu + b ρu πu

)+

(b Su .Du + b ρu εu

)+ ∇.b hu − ︷︸︸︷

h.n⊥�

with

ρu πu = (p∇.U)u − pu ∇.Uu,

ρu εu = (S.D)u − Su .Du ,

defining the SGS pressure dilatation and dissipation π and ε, respectively. Equatingthe two expressions, we have

∂b ρu eu

∂t+ ∇.b ρu euUu = − (

b ρu ∇.Uu + b ρu πu

) + (b Su .Du + b ρu εu

)+ ∇.b

(hu − bu

) +[

︷︸︸︷

ρeva − ︷︸︸︷

h.n⊥]�. (34)

The term in brackets is now the total energy contribution from the interface.

3.3. DERIVATION OF THE � EQUATION

The transport equation for the flame area per unit volume � has been derived forthe RANS case on several occasions [8, 27, 33]. A brief recap is given here usingthe conditional filtering methodology. We start by restating the definition of �,rewritten in (x⊥, η, ς)-coordinates as

� =∫D

G(x − x′)δ(x⊥ − xI )|J| dx⊥ d2x‖ (35)


Here |J| is the Jacobian for the surface: the Jacobian for the entire transformationbeing h⊥|J|. Candel and Poinsot give the transport equation for a surface elementδA as

dδA

dt= (−n⊥.∇UI .n⊥ + ∇.UI )δA. (36)

Our surface element

δA = |J| d2x‖

=∫

δ(x⊥ − xI )|J| dx⊥ d2x‖

=∫

δ(x⊥ − xI )1

h⊥dx′3.

If we multiply (36) by G and integrate over the surface, then the l.h.s. of theequation becomes∫

GdδA

dt= d�

dt+ ︷︸︸︷∇.UI� (37)

since the volume δV = dx′3 changes as the surface propagates. Hence (36) be-comes

d�

dt= ∂�

∂t+ (UI .∇)� = −︷︸︸︷

n⊥.∇UI .n⊥�. (38)

3.4. DERIVATION OF THE � EQUATION

A transport equation for the wrinkling factor introduced earlier can now be derived.Differentiating (31)

∂�

∂t= 1

|∇b |∂�

∂t− �

|∇b |∂|∇b |

∂t. (39)

Now, by differentiating |∇b | = nf .∇b ,

∂|∇b |∂t

= −︷︸︸︷

Ut .∇∇b − nf .∇︷︸︸︷

Ut .∇b − nf .∇Dc,

where︷︸︸︷

Ut is the surface-filtered effective velocity of the flame, defined so that

∂b

∂t+ ︷︸︸︷

Ut .∇b = 0. (40)


Since

∇[(

�

�

)2]

= ∇ (∇b .∇b) = 2(∇∇b ).∇b

= 2|∇b |�

(∇� − |∇b |∇�)

and so

∂|∇b |∂t

= −︷︸︸︷

Ut

�.(∇� − |∇b |∇�

) − nf .∇︷︸︸︷

Ut .∇b . (41)

Substituting this in Equation (39), and using Equation (38) to eliminate � we have

∂�

∂t+ ︷︸︸︷

U .∇� = −�︷︸︸︷

n⊥.∇UI .n⊥ + �nf .∇︷︸︸︷

Ut .nf

+ �(

︷︸︸︷

Ut − ︷︸︸︷

UI

).∇|∇b ||∇b | . (42)

The first term on the r.h.s. here can be further split into surface mean and fluctuatingcomponents

−�︷︸︸︷

n⊥.∇UI .n⊥ = 1

�n⊥∇︷︸︸︷

UI .n⊥ − Q, (43)

where Q involves surface fluctuation terms such as n ⊥ (Section 2) and represents

interaction between the interface and the turbulence leading to generation or re-

moval of interface area. The term n⊥∇︷︸︸︷

UI .n⊥ can be grouped conceptually withthe second term in (42), representing the effect of mean stretch and propagation on

�. The final term involves the difference between overall propagation velocity︷︸︸︷

Ut

and average interface velocity︷︸︸︷

UI , which increases with interface distortion. It alsoinvolves the second spatial derivative term ∇|∇b |/|∇b |. Evaluating this through ahypothetical Gaussian flame, this term tends to +∞ at the back of the flame and−∞ at the front. At the front this term forces � → 1, i.e. it has the effect ofsmoothing the interface, whilst at the back it has the reverse effect. Thus this termrepresents the process of cusp formation in the flame front.

4. Spherical Flame Front

A convenient example is the growth of a spherical flame front in stationary, isotropicand homogeneous turbulence. Homogeneous isotropic turbulence can be simulatedon a regular cubic grid of side 2n (here grids 64 cells on a side are used, i.e. 643

cells in total) using large scale forcing to generate turbulence [13]. The growthof a spherical flame front can be simulated in such a case, and comparisons with


experimental data have been presented for just such a case [23]. Here a recap isprovided of the detailed combustion modelling [37] (Section 4.1) and the resultscompared with theoretical expectations (Section 4.2). The case used is case A from[23]: an isooctane mixture at 1 atm, initial temperature 358 K, u′ = 2.36 m/s.The integral length scale L for this case has been measured at 20 mm, givinga turbulent Reynolds number based on this of ReT = 1700. The laminar flamespeed has been measured as Su = 0.434 m/s, and the laminar flame thicknessδl = 0.04 mm [5], giving a Damköhler number of 130, and a turbulent Karlovitznumber 0.3, indicating that the case falls easilly into the regime of laminar flameletcombustion [6].

4.1. DETAILED MODELLING

The final stage in formulating the combustion model is to provide models for thoseterms which cannot be explicitly expressed in terms of known quantities, or whichare too difficult to evaluate stabily or accurately. To close these equations, modelshave to be developed for the sub-grid scale (SGS) stress tensor, flux vectors anddissipation as well as for the filtered reaction rate. The SGS stress tensor and fluxvectors are not unique to reacting flows, and hence standard models may be applied.The principle difficulty in reacting LES is the proper treatment of the reaction zone;since the characteristic scales for the reaction processes are below the filter width,mean reaction rate models are required. Within our framework this is representedby the problem of solving equations for the geometric variables b and �.

Here we describe in brief the two-equation formulation of the Weller combus-tion model. Simpler models substituting algebraic equations for the various trans-port equations described here lead to alternative one-equation and algebraic for-mulations, further details being discussed in [37]. Rather than solve for both burntand unburnt phases separately we combine them and solve for phase-weightedquantities such as the total density

ρ = ρub + ρc(1 − b ). (44)

We can introduce a new regress variable b defined so that

ρb = ρub . (45)

Note that as b ∈ [0, 1], b ∈ [0, 1].Combining (31) and (45) with the b equation (27) gives

∂bρ

∂t+ ∇.(bρUu) = −ρuSu�|∇b |, (46)

where we have modelled the interface advance term︷︸︸︷

ρva in terms of the laminarflame speed Su and the unburned gas density ρu. The conditionally filtered unburntgas velocity Uu may be decomposed into the unconditioned density weighted ve-locity and the unburnt-burnt gas slip velocity thus Uu = U+(1−b)Uuc. By analogy


with the properties of laminar flames, the velocity difference across a turbulentflame is a function of the turbulent flame speed and the density ratio, in which caseperforming a simple one dimensional analysis of the turbulent flame brush thereresults

Uuc ≈(

ρu

ρc

− 1

)Su�nf . (47)

However, this does not correctly account for the distribution of �; the assump-tion being that � is constant through the flame, nor curvature effects or turbulentfluctuations in the phase velocities. Noting that the required transport form of theresulting equation must be the same as that of the rest of the filtered transportequations, a model for the turbulence effect is postulated, for consistency, as asimple gradient diffusion form, producing the following model for the slip velocity:

Uuc =(

ρu

ρc

− 1

)Su�nf − D

∇b

b(1 − b), (48)

where the turbulent diffusion coefficient D is the sub-grid diffusion coefficient.This models two types of effects. The second term is a gradient transport term,indicating transport proportional to ∇b. The first term however can act in theopposing direction and can thus represent the counter-gradient transport effectsconsidered important in most premixed and partially premixed combustion devices[34].

Combining Equations (46, 48) and rearanging yields

∂ρb

∂t+ ∇.(ρUb) − ∇.(ρD∇b)

= −∇ .b(1 − b)ρ

(ρu

ρc

− 1

)Su�nf − ρuSu�|∇b |. (49)

Some manipulation on Equations (44, 45) gives the result

ρu − ρ = ρ(1 − b)

(ρu

ρc

− 1

)(50)

Using this we can break down the first term on the r.h.s. of (49),

∇ .b(1 − b)ρ

(ρu

ρc

− 1

)Su�nf

= ∇ .b (ρu − ρ) Su�nf

= ∇.(b − b

)ρuSu�nf

= ρuSu�nf .∇ (b − b

) + (b − b

)∇.ρuSu�nf .


If we write nf .∇b � |∇b|, then (49) becomes

∂ρb

∂t+ ∇.(ρUb) − ∇.(ρD∇b)

= −ρuSu�|∇b| − (b − b

)∇.ρuSu�nf . (51)

In the final term on the r.h.s. here, gradients in ρu are likely to be negligable, andso we choose to ignore this term and use the equation

∂ρb

∂t+ ∇.(ρUb) − ∇.(ρD∇b) = −ρuSu�|∇b| (52)

as our modelled equation for b.We also need to model terms in Equation (42). The first two terms represent the

effects of strain and propagation respectively on the SGS wrinkling �. The thirdterm on the r.h.s. of Equation (42) represents the effect of differential propagationon the distribution of � through the flame, reducing generation at the front of theflame and enhanced generation at the back. This term involves high order deriva-tives which create numerical difficulties for LES. Instead of trying to model eachterm in detail we develop einsatz models which represent generation and removalof wrinkling, modelled by terms G� and R(� − 1). The problems associated withthese derivatives are thus avoided by including the effect directly into the modelfor G, resulting in the following simplified equation for �

∂�

∂t+ ︷︸︸︷

U s .∇� = G� − R(� − 1) + max[(σs − σt), 0]�. (53)

The final term here relates to the resolved strain rates

σt = 1

2‖∇︷︸︸︷

Ut + ∇︷︸︸︷

Ut

T ‖ and σs = 1

2‖∇︷︸︸︷

UI + ∇︷︸︸︷

UI

T ‖.The bounding here is introduced to account for the effects of lateral compressionon the flame surface, allowing only removal of wrinkling by resolved extensivestrain.

A spectral approach is applied to the modelling of the turbulence-flame in-teraction, in which the wrinkling of the flame is decomposed into a length-scalespectrum [34]. This approach lends itself naturally to LES in that sub-grid flameproperties may be obtained by integrating over the appropriate range of the spec-trum. However solution of the spectral evolution equations simultaneously with thetransport equation for � is prohibitively expensive and simple algebraic models areconsidered more appropriate. The current approach is based on the flame-speedcorrelation of Gülder [14], which has proved particularly good by comparison withfull spectral solutions, leading to

G = R�eq − 1

�eq

, R = 0.28

τη

�∗eq

�∗eq − 1

, (54)


�∗eq = 1 + 0.62

√u′

Su

Rη, �eq = 1 + 2(1 − b )(�∗eq − 1), (55)

where τη is the Kolmogorov time-scale, u′ is the sub-grid turbulence intensity andRη is the Kolmogorov Reynolds number. The surface filtered velocity of the flame︷︸︸︷

UI is modelled in a similar manner to the conditionally filtered unburnt gas velocity(see Equation (48)) as

︷︸︸︷

UI = U + (ρu/ρ − 1)Su�nf − ∇.(ρD∇b)

ρ|∇b| nf . (56)

The resolved strain-rate σt is obtained from U and the sub-grid turbulent flamespeed Su� by removing the dilatational component from the strain-rate in thedirection of propagation nf ,

σt = ∇.(U + Su�nf ) − nf .(∇(U + Su�nf )).nf . (57)

The gas expansion due to combustion is assumed to occur in the direction nf ,thus avoiding the need to accurately model the surface filtered gas velocity. Thesurface filtered resolved strain-rate σs is obtained similarly except that the effectsof flow field strain and propagation strain are separated in order that the influenceof flame wrinkling may be modelled appropriately; in the limit of very high wrin-kling (assuming isotropy) the compressive and extensive effect of the flow field areassumed to cancel, whereas the effects of propagation strain approaches half of thatfor wrinkling aligned with nf . The result is

σs = ∇.U − nf .(∇U).nf

�+ (� + 1)[∇.(Sunf ) − nf .(∇(Sunf )).nf ]

2�. (58)

The remaining requirement for closure is to provide a formulation for the lam-inar flame speed Su. This will depend heavilly on the physics of the case underconsideration. For some combustion problems it is reasonable to assume the lam-inar flame speed is unaffected by strain and curvature, and thus set Su = S0

u, theunstrained flame speed. However in most cases strain effects cannot be ignored[26]. For example, in the case of a flame trapped in the shear layer behind abackward-facing step [25], the strain effects near the step are important in re-ducing the effective reaction rate and preserving the Kelvin–Helmholtz instability.Certainly in the case under consideration both strain and curvature effects will beimportant [5]. One possible approach is to assume that the laminar flame speed is inlocal equilibrium with the local resolved strain rate σs (spectral modelling suggeststhat the effects of subgrid strain and curvature largely cancel), and thus the laminarflame speed can be set to the value attained when in equilibrium Su = S∞

u . Unfor-tunately, the chemical time scales of lean flames, such as lean propane flames, maybe comparable to the strain and transport time scales, in which case equilibriummay not be assumed and a full transport equation is required. By analogy with


the transport of the flame wrinkling, the filtered laminar flame speed is expected

to be transported at the surface filtered velocity of the flame︷︸︸︷

UI . Thus a transportequation of the form

∂Su

∂t+ ︷︸︸︷

U s .∇Su = −σsSu + σsS∞u

(S0u − Su)

(S0u − S∞

u )(59)

has been implemented. A detailed validation of this model is not possible withoutappropriate data on the effect of strain on laminar flames, however the above modeldoes seem to work well in practice.

4.2. RESULTS

Figure 3 shows the state of the b = 0.5 isosurface at t = 0.0054 s and t = 0.009 safter ignition. For direct experimental comparisons the exact definition of the ‘edgeof the flame’ will depend heavilly on the experimental technique used to analysethe flame front. For example, it is possible to realistically simulate the effect ofSchlieren photography by appropriate visualisation of the calculated data [32].However our purpose here is to be able to analyse the model, not make detailedcomparisons with experiment. The isosurface is coloured by the values of SGSwrinkling �. This enables us to examine the interaction between the grid scalewrinkling, as evidenced by the shape of the isosurface of b, and the subgrid scalewrinkling which is sampled by it.

Figure 4a shows a section through the computational domain at 0.009 s, show-ing contours of b ∈ [0, 1]. This should be compared with Figure 2b. The computa-tional mesh is shown overlaid for reference. It will be noticed that the filtered flamewidth is slightly broader, covering 3–4 cells rather than the 2–3 of the earlier result.This is not an enormous difference in breadth, and is attributable to numerical dif-fusion on a fairly coarse mesh. Convective terms in the b and b equations have beentreated using the gamma differencing scheme [17]: it may be that the error couldbe reduced somewhat using numerical methods designed specifically for interfacetracking [18]. Although conventionally the filter width for LES is linked to the celldimension this is not necessary, and the numerical broadening of the flame frontcould be interpreted as implying that the implicit filter for the indicator function isslightly wider than the grid spacing, possibly as a result of backscattering betweenthe flame and the turbulence. However the extensive investigation of the impact ofthe numerics on the SGS modelling is beyond the scope of this paper. Interestinglythe b isosurface is slightly narrower, although there is no reason to suspect this isanything other than a coincidence.

The modelling (e.g. Equation (57)) suggests that there should be a relationshipbetween � and the strain σs which could be extracted if the data is appropriatelyanalysed. σs is the flame surface strain field resulting from the surface filteringprocess, and � is also a flame surface property (although defined at all pointsin the flow). Figure 6a. plots values of � against D = 1/2‖∇U + ∇U T ‖ for


(a)

(b)

Figure 3. (a) Flame front as defined by b = 0.5, 0.009 s after ignition, (b) flame front0.017 s after ignition. The isosurfaces are shaded using values of the wrinkling � rangingfrom � = 1.0 (black) to � = 1.4 (white)


(a)

(b)

Figure 4. Flame front 0.009 s after ignition: (a) shows contours of b on a 2-d slice throughthe flame front, with the computational mesh superimposed; (b) shows contours of b.


Figure 5. Distributions of � (left column) and Su (right column) throughout the computa-tional domain at t = 0.005, 0.01, 0.015 s after ignition.

locations on the flame surface, at times t = 0.005, 0.01, 0.015 s after ignition. Aregression analysis demonstrates that there is a correlation between the two vari-ables with correlation coefficients around 0.56 for all three times. Extracting thisdata proved interesting in its own right. Initially the quantity m = |∇b| was usedas an indication of the location of the flame: values of this quantity significantlydifferent from zero (taken as values above the mean m) indicate a slope in b, inother words indicate the whole thickness of the flame front. Values of σ and �

from this region were then plotted. However this way only a very weak correlation


was found. However the location of the flame can be made much more precise,which was done by identifying cells where 0.49 < b < 0.51, and extracting thedata from these. This smaller sample is the one shown here, and produces a muchstronger correlation. A similar effect is noticeable in the correlation between Su andσ : for the mid-flame data shown in Figure 6b there is a good correlation (coefficient0.62), but with the dataset expanded to include the probable numerical tail of thedistribution the correlation is much weaker. Thus the modelling of the flame centre(values b ∼ 0.5) is physically accurate, but the flame leading and trailing edges areproblematic.

Figure 6c examines the correlation between the SGS and GS flame curvatures.� represents the SGS curvature, whilst b represents the GS geometry from whichit should be possible to extract GS curvature information. Since ∇b gives the GSsurface normal to the interface, a valid measure of its curvature will be ∇.∇b, soFigure 6c plots � vs. ∇2b for the flame centre, as determined above. The correlationis rather weak, as can be seen from the diagram. However the correlation betweenthe SGS turbulent kinetic energy of the fluid, k and �, plotted in Figure 6d is muchstronger (correlation coefficient of 0.7). This suggests that � is coupling stronglyto the SGS fluid flow and only weakly to the GS flow.

Figure 7 shows the evolution of various quantities relating to the combustionmodel as functions of time. Figure 7a compares the radius against time for severalrealisations of the same case with data from the Leeds bomb [4, 5]. A 323 box ofturbulence was used for this calculation, whilst another comparison for this caseusing a finer mesh (642) has been published elsewhere [23]. To suppliment this wepresent profiles of other important parameters. Figure 7b shows the variation offlame speed Su and surface wrinkling � with time. Both parameters are defined atall points in the mesh, so the spatial average over the whole domain is presented.However � in particular has little meaning away from the flame surface. Henceaverage values on the flame surface itself are also presented. This is achieved byfiltering these fields using the filter m = ‖∇b‖ iff m > m , m = 0 elsewhere (asdiscussed above). There is very little difference between the surface value and bulkvalue of Su, as one might expect. However there is a significant difference betweenthe value of � in the vicinity of the flame surface and that elsewhere. Figures 7c and7d demonstrate the variation of other bulk quantities, i.e. the SGS turbulent kineticenergy k and the enstrophy ζ = (1/2)|∇×U |, which is a measure of the vorticalnature of the flow. These are split into components averaged over the burnt andover the unburnt fraction of the gas, together with the total, thus indicating howthe turbulence is being modified by the passage of the flame front. Interestingly,the total turbulent kinetic energy increases with time, but the enstropy peaks andthen decreases. It seems likely that at about t = 10 ms the symmetry boundaryconditions begin to affect the propagation of the flame, causing the altered trendsgenerally after t = 10 ms.


(a)

(b)

(c)

(d)

Figure 6. Correlations between computed properties on the flame surface. (a) SGS wrinkling� vs. GS strain 1

2 ‖∇U + ∇U t‖. (b) Flame speed Su vs. (1/2)‖∇U + ∇U t‖. (c) � vs. GS

curvature ∇2b. (d) � vs. SGS turbulent kinetic energy k.


(a)

(b)

(c)

(d)

Figure 7. Variation with time of various quantities in the box: (a) tracks the progress of theflame surface, comparing Schlieren data from the Leeds bomb with simulated Schlieren radiievaluated from the computation; (b) shows variations of Su and flame wrinkling parameter �

throughout the box and at the flame surface; (c) shows the variation of the SGS turbulent ki-netic energy k inside and outside the flame surface, together with the total kinetic energy withinthe box; and finally (d) shows the same information for the enstrophy ζ = (1/2)|∇×U |.


5. Conclusions

This paper presents the mathematical and physical framework of a family of LEScombustion models for premixed and partially premixed combustion in the flameletregime. LES holds the prospect of significant improvements over traditional RANSturbulence modelling, because of its greater accuracy and ability to generate moreinformation to feed into the combustion modelling. In particular, such modelsshould be able to better treat countergradient transport, and simulate unsteadyfluctuations around a mean flow. In order to achieve our aim, we have to extendthe concept of conditional averaging to make it appropriate for LES. In RANS,conditional averaging is a technique allowing the decomposition of any fluid flowequations into equations appropriate for individual components of the flow. In do-ing so it provides a formal, rigorous framework for deriving the precise forms ofthe surface terms in the conditionally averaged equations. The extension of this toLES is relatively straightforward, and is in some ways more natural, since the LESformulation already incorporates the concept of a volume average. By introducingconditional and surface filtering operations a similar framework can be built up,with the effect of the interface surface on the bulk flow being dictated by the filtersupport. Although the aim of this paper is towards combustion modelling, thesemathematical techniques should prove useful in modelling other multicomponentflows. Various issues relating to the filtering of an appropriate indicator functionare discussed. In particular it is shown by the simple procedure of filtering a 2-dstep function that the filtered indicator function is spread over 2–3 cells. Thus theproblem of performing LES of the flame front in the high Damköhler regime iscircumvented by simulating the filtered indicator function (representing large scalegeometric information) and modelling the small scale geometric information.

Within this framework filtered versions of the NSE are derived. The resultingadditional interface terms need to be modelled, and from this a family of LEScombustion models of varying complexity and accuracy can be derived. Appro-priate models based on the flame front wrinkling have been introduced and testedin previous papers. The modelling is analogous to that used for the RANS variantof these models, with only the interpretations of the variables being different. Ifanything the detailed modelling is more appropriate for the LES case, with thescale division of the flame front geometry described above. The ability of suchmodels to simulate complex flows has already been demonstrated [37] for com-bustion behind a backward facing step. This test case is strongly time-dependent,and would be a real problem for a RANS-based combustion model. The case usedin this paper is simpler, and the Weller models can be shown to work well forthis case as well, as comparison with experimental cases has shown [23]. Sincethe bias of this paper is towards the theoretical aspects of the model, the sphericalflame case is used merely for illustrative purposes, readers being directed to theearlier paper for details of the comparison. It is seen that the filtered indicator


function as calculated is in close qualitative agreement with that generated bydirect filtering of the indicator function. Taken as a whole therefore, the WellerLES combustion models are both accurate and well grounded in theory.

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Large Eddy Simulation of Premixed Turbulent Combustion Using E Flame Surface Wrinkling Model