LES combustion modeling using a multi-phase analogy

The reactive flow equations are the balance equations of mass, momentum and energy describing convection, diffusion and chemical reactions, [4-6, 10]. In LES, [2], all variables are decomposed into resolved and unresolved (subgrid) components by a

spatial filter such that

f = ~ f + f ′′

, where

ρ ρ /

~ f

f =

is the Favre filtered variable. Filtering the reactive flow equations yields,

⎪⎪

⎪

⎩

⎪⎪

⎪

⎨

⎧

Σ + +

−

⋅

∇ +

⋅

∇

+ +

∇

⋅

=

⋅

∇ +

−

⋅

∇ +

−∇

=

⋅

∇ +

+

−

⋅

∇

=

⋅

∇ +

=

⋅

∇ +

=

⊗

).

~ ( )

~ (

)

~ (

~ ) ( ~

~ ) (

), (

~) ( ~

~) (

, ) (

~) ( ~

~) (

, 0

~) ( ) (

,

1 θ

σ ρ

∂ ρ

ρ

∂ ρ ρ

∂

ρ ρ

∂

ρ ρ

∂

i i f N h i

t s

s t t

i i i i

i t t

h w p

p h

h

p

w Y

Y

&

&

b h v

v S v

B S v

v v

b j v

v

(1) Here,

ρ

is the density, v the velocity, p the

pressure, S the viscous stress tensor,

∫

Σ

= ^T

T pi i i

s Y C dT

h 0 , the sensible enthalpy, T the temperature, h the heat flux vector, Yi the species mass-fraction,

w

_&i the species reaction rate, ji the species mass-flux, θ

i

hf_, the species formation enthalpies and

σ

the radiative heat loss which will be neglected. The subgrid flow physics is concealed in the subgrid stress tensor _B=ρ(_v

~

_⊗v−~_v⊗~_v) _and

flux vectors (

~

~~)

i i

i vY vY

b =ρ − and

~) (

~

_h ~_h

h v v

b =ρ − , which results from filtering the convective terms. Following [6] we postulate that the gas mixture behaves as a linear viscous fluid with Fourier heat conduction and Fickian species diffusion so that j_i≈D_i∇Y~_i, p≈ρRT~,

DD

S ≈2μ~ , h≈κ∇T~, respectively. Here, i

i Sc

D =μ/ is the species diffusivities, R the composition dependent gas constant,

μ

the viscosity, D~ ( v~ v~ ) ( ~v)I

3 1 2

1 ∇ +∇ − ∇⋅

= ^T

D the

deviatoric part of the rate of strain tensor,

Pr μ /

κ =

the thermal diffusivity, in which Sci and Pr are the Schmidt and Prandtl numbers, respectively. Finally, the filtered reactions rates are defined as

w

_&i

= M

_i

Σ

^M_j=1

P

_ij

w

_&j , in which Mi is the molar mass of specie i, Pij the stoichiometric matrix

and ^{ij j ij}

N

i P

i N j i P

j T k Y

w = ^Σ=1 β Π₌₁

& ρ the reaction rate

of the j^th reaction step, with kj being the Arrhenius rate.

2.1 Subgrid flow modelling

To close the LES equations (1) we need to provide models for B, bi, bE and w&_i , and considering first the subgrid stress tensor and flux vectors, these are

not unique to reactive flows and closure models can be acquired from the plethora of subgrid models for non-reactive flows, [2-3]. Here, we use the Mixed

Model (MM), [33], with D

kD v

v v v

B_=ρ(~

~

_⊗~₋~~_⊗~~)₋2_μ ~ _, Sc i

i i

i Y Y Y

t

k ~

~~)

~ ~~

(~

~

_{− − ∇}

=ρ v v μ

b and

s s

s

h h h h

t

k ~

~~ )

~ ~~

(~

~

_{− −Pr ∇}

= ρ v v ^μ

b , in which the subgrid

viscosity,

μ

_k

= c

_k

Δ k

^1/2, is provided by an equation for the subgrid kinetic energy, k, [33], and where Sct and Prt are the turbulent Schmidt and Prandtl numbers. The model coefficients are calculated under the assumption of an infinite inertial sub-range, [2]. To reduce the computational cost we use wall-modeled LES, in which a model is used to handle the near-wall flow physics, [34]. This model is based on replacing the viscosity (

μ + μ

_k), and thermal and species diffusivities (μ/Pr+μ_k/Pr_T) and (μ/Sc_i+μ_k/Sc_T ), respectively) in the first grid point at the wall, with values consistent with the logarithmic law of the wall.

2.2 A multi-phase framework for subgrid combustion modelling

Considering next the filtered reaction rates w&_i ^, incorporating the turbulence chemistry interactions, being notoriously difficult to model because of their non-linear nature, [1]. Using the phenomenological models, [16-17, 35], based on experimental data, and DNS results, [19-20], a cartoon of turbulent mixing and combustion may be drawn. In this, turbulent reacting flows may be viewed as a muddle of vortex structures of different topological character, sheets, ribbons and tubes, in which the tubes and ribbons carry most of the high-intensity vorticity and dissipation. This implies that the fine-structure regions, denoted by (*), are embedded in a surrounding fluid, here denoted by (⁰), will be responsible for most of the molecular mixing, chemical reactions (if the temperature is sufficiently high) and heat release. This cartoon is similar to that of the Eddy Dissipation Concept (EDC), [13, 21, 36], but here a different strategy, based on an analogy with modeling multiphase flows, will be used to model the filtered reaction rates w&_i ^.

For the derivation of the proposed model we let Ψ=[Y_i,h_s] be the composition space that

satisfies the equation i

i i

i

t ρψ ρ ψ ω

∂ ( )+∇⋅( v )=∇⋅k + _& , in which

i i =Y

ψ , k_i= j_i and

ω

&_i

=

w&_i for 1<i<N, where N denotes the number of species in the species

equation (13) whereas

ψ

_N+1

=

h_s,

k

_N+1

= h

, and )

~ (

~ 1 ,

1

∂

θ

ω

&_N+ =S⋅∇v+ _tp+∇p⋅v+Σ_i^N= w&_ih_fi

are the terms associated with the energy equation (14). As for the theory of multi-phase flows, e.g.

[37], a phase indicator function, denoted I_α, with

=1

α

in the fine structures (*) and

α

=0 in the surroundings (⁰) is next introduced to differentiate between these regions so that the local balance equations for the composition space becomes,

, )

(

) (

) (

, ,

,

, ,

α α α α α

α α α α

α α

ω ψ ρ ψ

ρ

∂

i i

i

i i

t

I I

I I

Μ + +

⋅

∇

=

⋅

∇ +

&

k

v (2)

in which

Μ

_i,α denote the exchange terms at the immaterial surface separating the fine structures and

=0

α

in the surroundings. By summing over

α

in (2), provided that the exchange terms satisfies

0 ) ( _,

1

0 Μ =

Σ_{α= iα} , the composition space equations are recovered. Although achievable, velocity differences between the fine-structures and surroundings are not included. In the framework of LES the species mass fraction equations are filtered over space to obtain equations for the large energetic scales of motion. If we take the filtering to correspond to a box-filter, covering the cell volume,

∆V, as often done in LES, [2], the volume fraction of phase

α

is,

, 1 with

/

) , ( 1 0 1

= Σ

= Δ Δ

=

=

= Δ ∫Δ α α

α α

α α

γ

γ I tdV V V I

V V x

(3)

so that for products

I

_α

ψ

_i,α and

ρ

_α

ψ

_i,α

I

_α it can easily be shown that (

ψ

_i,α)_α =

ψ

_i,αI_α/I_α and

α =

α α

ψ

ρ

_i, I

γ

_α(

ρ

_α

ψ

_i,α)_α =

γ

_α(

ρ

_α)_α(

ψ

~_i,α)_α

. Filtering the equations (2) then yields,

, )

( ) )

( (

~)

~ ) ( ) ( ( )

~ ) ( ) ( (

, ,

,

, ,

α α α α α

α α α

α α α α α α

α α α α

ω γ γ

γ

ψ ρ γ ψ

ρ γ

∂

i i

i i

i i

t

Μ + +

−

⋅

∇

=

⋅

∇ +

&

b k

v

(4) in which

γ

_αb_i =

ρ

_α

ψ

_i,αI_αv−

γ

_α(

ρ

_α)_α(

ψ

~_i,α)_αv~

denotes the subgrid transport term, which will typically be modeled by a conventional subgrid viscosity model. Summing over the filtered equations (4) results in that the filtered exchange terms satisfy Σ¹_α=0(Μ_i,α)=0 so that the LES composition space equations (12) and (14) are recovered. Returning next to the old notions, with (*) denoting the fine structures and (0) the surroundings,

such that

ψ

_i^*

= ( ψ ~

_i,1

)

₁ and

ψ

_i⁰ =(

ψ

~_i,2)₂, and

= +

=

^{* * 0 0}

~

i i

i

γ ψ γ ψ

ψ γ

^*

ψ

_i^*

+ ( 1 − γ

^*

) ψ

_i⁰, we then obtain the set of equations,

⎪⎪

⎪

⎩

⎪⎪

⎪

⎨

⎧

Μ + +

−

⋅

∇

=

⋅

∇ +

Μ + +

−

⋅

∇

=

⋅

∇ +

. ))

( (

~) (

) (

, ))

( (

~) (

) (

0 0 0 0

0 0 0 0

0 0

*

*

*

*

*

*

*

*

*

*

i i i

i

i t i

i i i

i

i i

t

ω γ γ

ψ ρ γ ψ

ρ γ

∂

ω γ γ

ψ ρ γ ψ

ρ γ

∂

&

&

b k

v b

k

v

(5)

The above equations can be solved together for ψ_i^*

and ψ_i⁰ after which the LES quantity

ψ

^~_i can be constructed from

~

^{* *}

( 1

^*

)

⁰

i i

i

γ ψ γ ψ

ψ = + −

once

the fine structure volume fraction,

γ

^*, is known. A more convenient and straightforward approach is however to solve the balance equations for the fine structure fractions,

ψ

_i^*, (51) together with the LES balance equations for

ψ

^~_i, or more precise-ly the LES species equations (12) and energy equation (14).

Noticing that ω&_i=γ^*ω&_i^*+(1−γ^*)ω&_i⁰, and more specifically that w_&i=γ^*w_&i^*+(1−γ^*)w_&i⁰, and taking advantage of the theoretical, [16-17], experimental, [35], and computational, [20], observations that (fast exothermal) reactions mainly occur in the fine structures, (12) becomes,

, )

(

~) ( ~

~)

( _{i i i i} ^* _i^*

t ρY ρY γ w_&

∂ +∇⋅ v =∇⋅ j −b + (6) emphasizing the importance of the volume fraction of the fine structures. Equation (6) further implies that the reaction rate can be expressed as

* 0

0 _{* *}

* (1 )

) ( )

( _{i i i i}

i w d w w w

w& =∫_ψ℘^Ψ & ^{Ψ Ψ}≈γ & + −γ & ≈γ &

, in which ℘(Ψ) is hence a probability density

function of the form )

( ) 1 ( ) ( )

(Ψ = ^* Ψ−Ψ^* + − ^* Ψ−Ψ⁰

℘ γ δ γ δ .

Moreover, by summing over the N species equations in (5) and taking specifically into account that

1 ) ( )

( ₁ ⁰

1 ^* =Σ =

Σ_i^N₌ ψ_{i i}^N₌ ψ_i and

0 ) ( )

(

^* ₁ ⁰

1

= Σ =

Σ

_i^N₌

ω _&

_i ^N_i=

ω _&

_i , it immediately follows

that,

), (

~) ( ) (

and

) (

~) ( ) (

1 0 0 0

0 0

1 *

*

*

*

*

i N t i

N i t i

Μ Σ

=

⋅

∇ +

Μ Σ

=

⋅

∇ +

=

= v v ρ γ ρ

γ

∂

ρ γ ρ

γ

∂

(7)

the sum of which must satisfy the continuity equation (11). This constraint subsequently results in

that

m & = Σ

_i

( Μ

_i^*

) = − Σ

_i

( Μ

_i⁰

)

, where we denoted

by m& the exchange rate of mass between the fine

structures and the surroundings. This exchange rate of mass between the fine structures and surroundings is a key quantity for accurate predictions of species mixing and reactions.

2.3 Subgrid combustion modelling

To provide a closure of the exchange rate of mass,

m &

, through the immaterial surface separating the

fine structures and surroundings, we suppose that if there is a dynamic equilibrium state between the fine structures and the surroundings, which is characterized by the value

γ

_eq^* (which is to be discussed later), then mass exchange rate between the fine structures and the surroundings is equal to zero, i.e. m_&(

γ

_eq^* )=0. For this equilibrium state, the surface separating the surroundings and fine structures can be considered material. Developing

m &

in the vicinity of

γ

_eq^* , whilst retaining only the

first-order linear term, assumed proportional to )

(γ^*−γ_eq^* , results in that, , / ) (γ^* γ^* τ^*

ρ _eq

m& =− − (8)

where

τ

^* is the fine structure residence time, and where the term

ρ / τ

^* follows from dimensional considerations. The hypothesis underlying (8) refers to high Re-number flows, in which combustion takes place in concentrated regions, whose entire volume is a small fraction of the total volume. The application of this approach for lower Re-numbers, i.e. in the flamelet regime, is not straightforward. We conclude from (8), that if m& >0^, γ^*<γeq^* , then the exchange rate of mass is directed from the surroundings to the fine structures. Otherwise, if

<0

m& , γ^*>γ_eq^* , then the exchange rate of mass is

directed from the fine structures to the surroundings.

We also note, that this mass exchange rate is due to convection through the interface separating fine structures and surroundings. Using the closure model (8),

γ

^* can be obtained from (71),

. / ) (

~) ( )

(γ^*ρ^* γ^*ρ^* ρ γ^* γ^* τ^*

∂_t +∇⋅ v =− − _eq

(9) Next we consider the necessary modeling of the

exchange terms

Μ

_i^* and

Μ

_i⁰ in equation (5). As follows from the core physical considerations,

Μ

_i^*

and

Μ

_i⁰ contain two kinds of terms. The first type of term, here denoted by

Θ

_i^* and

Θ

_i⁰, is due to the exchange rate of mass between the fine structures and the surroundings (through the surface separating the fine structures and surroundings) as discussed above. If the exchange rate of mass is absent (i.e. in the dynamic equilibrium state, when the surface separating the fine structures and surroundings can be considered as material one),

m & = 0 ,

as there is no transport through the interface between the fine structures and the surroundings, these two terms both turn to zero. The second type of term, here denoted by

Ω

^*_i and

Ω

_i⁰, is due to molecular diffusion through the interface between the fine structures and surroundings. Indeed, even if the exchange rate of mass is absent (i.e. in dynamic equilibrium), i.e.

,

= 0

m &

there is exchange through the interface due

to molecular diffusion. If

ψ

_i^*

= ψ

_i⁰, these terms however turn to zero, since in this case there is no molecular diffusion. Based on the above physical considerations we propose to model these exchange terms as,

, ,

) ( ) (

, / ) ( ,

/ ) (

, ,

* 0 2

0 1 2

* 1

0

* * 0 0

* *

0 0 0

*

*

*

*

* *

* *

i i i i

i

i i i

i i

i i

i i i i

i i

m m m

m+ + − Θ =−Θ

= Θ

−

−

= Ω

−

= Ω

−

−

= Ω

Ω + Θ

= Μ Ω

+ Θ

= Μ

ψ ψ

τ ψ ψ ρ γ τ

ψ ψ ρ γ

&

&

&

&

(10) thereby satisfying Σ¹_α=0(Μ_i,α)=0 (in fact,

separately for both two types of terms,

0 ) ( _,

1

0 Θ =

Σ_{α= iα} and Σ¹_α=0(Ω_i,α)=0, as has to be). We note, that in correspondence with the underlying physics, if mass exchange is directed from the surroundings to the fine structures, m& >0^,

one obtain from (10) that

Θ

_i^*

= m _& ψ

_i⁰ and

0

* 0

i i

i

= − Θ = − m _& ψ

Θ

. Similarly, if the mass

exchange is directed from the fine structures to the surroundings, i.e. for m& <0, one has that

*

*

i

i

= m _& ψ

Θ

and

Θ

_i⁰

= − Θ

_i^*

= − m _& ψ

_i^*.

By assuming that the convective and diffusive terms in (51) can be neglected we have that

*

0

*

*

ω

_i

+ Μ

_i

=

γ _&

. Summing in this relation over

i∈[1,…, N], and taking into account that

0 )

(

^*

=

Σ

_i

ω _&

_i , as follows from the law of mass

action, it follows that

Σ

_i

( Μ

_i^*

) = 0

, which, by the definition of

m &

^as

m & = Σ

_i

( Μ

_i^*

)

, results in that

=0

m& . By using the linear model introduced in (8)

this results in that

γ

^* =

γ

_eq^* . The fact that m& =0

when convection and diffusion can be neglected at

the subgrid level in (51) further implies that

*

= 0

Θ

_i , which in turn implies that

*

*

) /

(

⁰

* *

* _i

γ ρ ψ

_i

ψ

_i

τ

i

= Ω = − −

Μ

. From the

relation

γ

^*

ω

_&i^*+Μ_i^* =0 it then follows that

*

) /

*

(

⁰

*

ρ ψ ψ τ

ω &

_i

=

_i

−

_i , which may be further

rearranged, using the aforementioned definition 0

*

* (1 )

~ ^*

i i

i γ ψ γ ψ

ψ = + − , to the more appropriate form,

, ) 1 /(

/

~ )

(ψ_i^*−ψ_i τ^* −γ^* =w&_i^*

ρ (11)

which can be recognized as the standard LES-PaSR model, [12]. Note also that the EDC model, [13, 21], bears some resemblance of the algebraic model (11).

Equation (11) can be arrived at and understood also from simple mass balance considerations as done in the conventional PaSR and EDC models but is here (i) put on a more solid mathematical foundation and (ii) extended to non-equilibrium situations prevailing in some combustion applications.

The framework developed so far can be used to provide a more rigorous foundation for the LES-PaSR combustion model, [12], and to propose an extension of this model, referred to as the Extended PaSR (EPaSR) LES model. In the LES- PaSR model, [12], the continuity, momentum and energy equations, (11), (13) and (14), respectively, are solved together with the species equati-on (6), in which the filtered reaction rates are modeled by

) ( ^*

*

*

* _{i i} Ψ

i w w

w& =γ & =γ & , in which

ψ

^*_i results from

solving (11) in which

γ

^* is provided by

γ

_eq^* , yet to be specified. The PaSR-LES model has been proven reliable and accurate, [12, 38], but a better model is needed for more complex flows. In the proposed LES-EPaSR model the continuity, momentum and energy equations, (11), (13) and (14), respectively, are solved simultaneously with the species equation (6) in which w&_i =γ^*w&_i(Ψ^*), where

ψ

_i^* results from solving the transport equations (51) with source terms M_i^*, provided by (101) and γ^* provided by solving (9). To avoid calculating the fine-structure density,

ρ

^*, equations (51) and (9) will here be simplified by replacing

ρ

^* by

ρ

so that in the LES-EPaSR model the fine structure composition and enthalpy (or temperature) is given by,

⎪⎪

⎩

⎪⎪⎨

⎧

−

−

=

⋅

∇ +

−

− −

+ +

−

⋅

∇

=

⋅

∇ +

, / ) (

~) ( ) (

, ) 1 /(

/

~ ) (

)) (

(

~) (

) (

*

* *

*

*

*

*

* *

*

*

*

*

*

*

*

*

τ γ γ ρ γ

ρ γ

ρ

∂

τ γ ψ ψ

ρ γ ψ

ω γ γ

ψ γ ρ ψ

γ ρ

∂

eq t

i i i

i i

i i

i t

m v

b k v

&

&

(12)

for 1≤i≤N+1, in which ~ ^{* *} (1 ^*) ⁰ i i

i γ ψ γ ψ

ψ = + − has

been used to simplify the rhs of (121) and in which / *

) (γ^* γ^* τ

ρ _eq

m_& =− − . To close (12) the value of

*

γ

eq is required which will be discussed next.

2.4 Modeling the subgrid time scale and the equilibrium reacting volume fraction

To finally close the EPaSR and PaSR models, the subgrid time,

τ

^*, and reacting volume fraction,

γ

^*, needs to be provided. In what follows we propose sub-models for both these quantities based on the physical description provided in Section 2, and the references cited.

To estimate the subgrid time,

τ

^*, we first observe that the fine scale structures in a volume,

V

*

Δ

, are generally anisotropic (sheets, ribbons and tubes) and influenced by the subgrid velocity stretch,

′/Δ

v , the characteristic time of which is Δ/v′. The smallest quasi-equilibrium scale of the fine structures, l_D, is controlled by molecular viscosity and the characteristic time of the stretch Δ/v ′ , such that _lD =(

ν

Δ/v′)^1/2. Its similarity with the Taylor length scale, l_T =(

ν

l_I /v′)^1/2, defined from the molecular viscosity and the integral time scale l_I /v′, suggests that l_D is the dissipative length scale of the smallest resolved scales. We further assume that area-volume ratio,

Δ S

^*

/ Δ V

^*, is solely determined by l_D ^{so that}

V D

S^*/Δ ^* ≈1/l

Δ . For small deviations from the equilibrium state, a simple relaxation equation can be formulated for the variation of

Δ V

^* so that,

), (

) / ( /

)

( V^* dt v_K S^* V^*_eq V^* V_eq^*

d Δ =− Δ Δ Δ −Δ

(13) in which we take into account that the change of the

volume

Δ V

^* is due to the propagation of the surface

Δ S

^* with relative velocity on the order of the Kolmogorov velocity, v_K, [27]. From (13) it also becomes evident that the subgrid time scale may be expressed as,

. / )

) / (

( ^{* * 1}

*= vK ΔS ΔV eq ⁻ =lD vK

τ (14)

Note also that (13) and (14) degenerates into (9) when divided by ∆V. It should however also be noted that if it assumed that area-volume density,

*

*

/ V

S Δ

Δ

, is instead determined by the

Kolmogorov scale, l_K, we obtain

τ

^*=

τ

_K, which contradicts the physical description of the fine structures discussed above. Using the definition of the Kolmogorov length- and time scales,

4 / 1 3/ ) (

ν ε

K =

l ^and

τ

^K =l^K /v^K, respectively, this finally results in that

τ

^*=

τ

_K

τ

_Δ in which

Δ v′

Δ = /

τ

is the time of the subgrid velocity stretch. This represents the geometrical mean of the (short) Kolmogorov time and the (long) time scale associated with the subgrid velocity stretch.

To model the equilibrium reacting fine- structure fraction,

γ

_eq^* , we use dimensional conside- rations and analysis of two limiting cases (fast and slow chemistry in comparison with mixing). More precisely, following from the dimensional considera- tions we write,

).

/ , / , /

* (

K K m c

eq= f τ τ τ_Δ τ Δ l

γ (15)

Since we consider the equilibrium state (m& =0^),

the parameters that characterize unsteady and convective effects are not present in (15). Let us for simplicity denote the characteristic time and length scales ratios by

x = τ

_c

/ τ

^*, y=

τ

_Δ/

τ

_K and

z =Δ/lK, respectively, and by neglecting intermittency we have that y=z^2/3. Thus we consider further that

γ

_eq^* depends solely of x and y,

).

,

* (

y x

eq = f

γ

(16)

There are two limiting cases for the parameter y, namely y>>1 and y≈1. The first case is indeed LES since from above it follows that Δ>>l_K, whereas the second case is indeed DNS since all scales are resolved, i.e.

τ

_Δ ≈

τ

_K and Δ≈l_K , and thus

1 ) ,

* = f(x y =

γ

eq irrespectively of x. For the first case we consider two limiting situations: (i) x<<1 (or

τ

*

τ

_c

<<

), which corresponds to fast chemistry, in which combustion is limited by mixing, and (ii) x>>1 (or

τ

_c

>> τ

^*), which corresponds to slow chemistry, in which combustion is limited by the chemical reaction rate. In the first situation (x<<1 or

τ

*

τ

_c

<<

), when combustion is limited by mixing, the product

γ

_eq^*

ω

_&i^* has to be of the order

1 / τ

^*, and if we take into account that

ω

&_i^* ∝1/

τ

_c^,

. 1 1 , ) ,

* = f(x y ≈x x<< and y>>

γeq (17)

In the second situation (x>>1 or τ >>_c τ^*), when combustion is limited by the reaction rate, the prod-

uct

γ

_eq^*

ω

_&i^* has to be of the order 1/τ_c. Thus, if we

take into account that

ω

&_i^* ∝1/

τ

_c^,

. 1 , 1 , 1 ) ,

* = f(x y ≈ x>> y>>

γeq (18)

In order to provide a reliable model for

γ

_eq^* , respecting the underlying cases and situations discussed above we propose to model

γ

_eq^* in the following manner,

), / ( )]

/(

| )) / ( 1 (

) ( )]

1 /(

[ )) ( 1 ( ) , (

*

*

K c

c K eq

F F

y F x x y F y

x f

τ τ τ τ τ τ τ γ

Δ

Δ + +

−

= +

+

−

=

=

(19) where )F(y is a monotonous function satisfying

0 ) 1 ( =

F and F(∞)=1. In this paper solely the case

>>1

y , (τ >>_Δ τ_K) is considered resulting in that (19) reduces to,

.) /(

) 1 /(

) ,

( ^*

* τ τ τ

γ_eq = f x y =x +x = _{c c}+ (20) This model for the reacting volume fraction is identical to the phenomenologically derived mod-el successfully used in previous LES, in which the chemical time scale is estimated as

τ

_c ≈

δ

_u /s_u ≈

/ s

_u2

ν

and the subgrid time scale is estimated as τK

τ

τ^*= _Δ . Introducing τ_Δ =Δ/v′ and 2

/ )1 / (ν ε

τ =_K , in which ε=(v′)³/Δ, (20) becomes )γ_eq^* =(ν^3/4v′^5/4)/(s_u²Δ^3/4+ν^3/4v′^5/4 . 3. Global hydrocarbon combustion chemistry

Accurate predictions of temperatures and species concentrations require the use of carefully selected reduced schemes, and here reduced three-step mechanisms for CnHm-air combustion are used. The mechanism for C3H8 are detailed in Table 1, for which the Arrhenius coefficients have been optimiz- ed to match the laminar flame speeds and tempera- tures of the Gri-Mech 3.0, in lean premixed flames, [40]. The third reaction step, modeling NO formation, is the sum of two rates, one for thermal NO (only dependent on temperature) and one for prompt NO (depending on the hydrocarbon fuel and H2O). Most NOX reacts to NO2 after combustion, however, close to the flame, NO is dominant, and

therefore, it is sufficient to limit the mechanism to NO. Associated with the reaction mechanism is the treatment of molecular diffusion, that is here simplified and based on species Schmidt numbers so that j_i ≈(μ/Sc_i)∇Y~_i, where the values of the individual species Schmidt numbers are listed in Table 2. For a critical evaluation and discussion of this approach to model the molecular diffusion we refer to Giacomazzi et al, [41].

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