Arne Scholtissek
email: scholtissek@stfs.tu-darmstadt.de

Sebastian Popp

Sandra Hartl

Hernan Olguin

Pascale Domingo

L Vervisch

Christian Hasse

Derivation and analysis of two-dimensional composition space equations for multi-regime combustion using orthogonal coordinates

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The distinction between the premixed and non-premixed combustion regime has been essential in the development of combustion theory and modeling. However, practical combustion systems often exhibit, sometimes by design, complex flame structures which cannot be characterized as purely premixed or purely non-premixed [START_REF] Knudsen | Capabilities and limitations of multi-regime flamelet combustion models[END_REF][START_REF] Masri | Partial premixing and stratification in turbulent flames[END_REF][START_REF] Hartl | Assessing the relative importance of flame regimes in Raman/Rayleigh line measurements of turbulent lifted flames[END_REF][START_REF] Butz | Local flame structure analysis in turbulent CH4/air flames with multi-regime characteristics[END_REF]. Instead, effects such as partial premixing, mixture inhomogeneities/stratification, hot product recirculation, entrainment of oxidizer, and local flame extinction can lead to mixed combustion regimes which are denoted as multi-regime combustion in this work. For the last decade, multi-regime combustion has been a topical research area resulting in a broad range of experimental and numerical works as reviewed by Masri [START_REF] Masri | Partial premixing and stratification in turbulent flames[END_REF]. To date, modeling of multi-regime combustion still poses a major challenge since many combustion models employed in CFD simulations of practical combustors rely on restrictive assumptions regarding the structure of reaction zones.

The identification of combustion regimes with respect to premixed, non-premixed or multiregime characteristics is non-trivial and multiple works exist regarding this topic. Yamashita et al. [START_REF] Yamashita | A numerical study on flame stability at the transition point of jet diffusion flames[END_REF] suggested the flame index, a scalar which considers the alignment of fuel and oxidizer gradients. The flame index characterizes a flame as premixed if the scalar product of the local fuel and oxidizer gradient is positive and as non-premixed if it is negative. This criterion was later used for analyzing well-resolved numerical data and also extended by other groups [START_REF] Domingo | Partially premixed flamelets in LES of nonpremixed turbulent combustion[END_REF][START_REF] Mizobuchi | A numerical analysis of the structure of a turbulent hydrogen jet lifted flame[END_REF][START_REF] Domingo | DNS analysis of partially premixed combustion in spray and gaseous turbulent flame-bases stabilized in hot air[END_REF][START_REF] Mizobuchi | A numerical study on the formation of diffusion flame islands in a turbulent hydrogen jet lifted flame[END_REF][START_REF] Fiorina | Approximating the chemical structure of partially premixed and diffusion counterflow flames using FPI flamelet tabulation[END_REF]. Another combustion regime indicator was proposed by Knudsen and Pitsch [START_REF] Knudsen | A general flamelet transformation useful for distinguishing between premixed and non-premixed modes of combustion[END_REF] who assessed flame structures based on a balance of the local chemical source term, molecular mixing, and unsteady effects. In comparison to the flame index, their combustion regime index showed comparable predictions, but differences were found for regions where premixed and non-premixed flame structures occur in close proximity. While the aforementioned combustion regime identifiers rely on scalar gradient information, which is only accessible from well-resolved numerical simulations, Hartl et al. [START_REF] Hartl | Regime identification from Raman/Rayleigh line measurements in partially premixed flames[END_REF] developed a gradient free regime identification (GFRI) approach. The method can also be applied to experimental data and allows distinguishing combustion regimes based on the local thermochemical state obtained from Raman/Rayleigh spectroscopy. Since, the GFRI approach has been refined to identify the relative importance of combustion regimes [START_REF] Hartl | Assessing the relative importance of flame regimes in Raman/Rayleigh line measurements of turbulent lifted flames[END_REF] and to differentiate local combustion regime characteristics [START_REF] Butz | Local flame structure analysis in turbulent CH4/air flames with multi-regime characteristics[END_REF].

Besides the analytical tools for combustion regime identification, different modeling approaches exist to describe multi-regime combustion characteristics. Here, a composition space modeling approach is used and related previous works are briefly revisited in the following. Well-established composition space models, such as the classical non-premixed flamelet model [START_REF] Peters | Laminar flamelet concepts in turbulent combustion[END_REF], Flamelet Generated Manifolds (FGM) [START_REF] Van Oijen | State-of-the-art in premixed combustion modeling using flamelet generated manifolds[END_REF] and Flame Prolongation of ILDM (FPI) [START_REF] Gicquel | Laminar premixed hydrogen/air counterflow flame simulations using flame prolongation of ILDM with differential diffusion[END_REF] rely on a one-dimensional representation of the reaction layer (non-premixed or premixed, respectively). It has been shown that these modeling approaches can capture multi-regime/partially premixed combustion characteristics to a certain extent [START_REF] Domingo | Partially premixed flamelets in LES of nonpremixed turbulent combustion[END_REF][START_REF] Domingo | DNS analysis of partially premixed combustion in spray and gaseous turbulent flame-bases stabilized in hot air[END_REF][START_REF] Fiorina | Approximating the chemical structure of partially premixed and diffusion counterflow flames using FPI flamelet tabulation[END_REF][START_REF] Van Oijen | State-of-the-art in premixed combustion modeling using flamelet generated manifolds[END_REF][START_REF] Fiorina | Modelling non-adiabatic partially premixed flames using flame-prolongation of ILDM[END_REF][START_REF] Cònsul | Analysis of the flamelet concept in the numerical simulation of laminar partially premixed flames[END_REF][START_REF] Popp | LES flameletprogress variable modeling and measurements of a turbulent partially-premixed dimethyl ether jet flame[END_REF][START_REF] Han | The role of tangential diffusion in evaluating the performance of flamelet models[END_REF]. However, as stated initially, these models rely on restrictive a-priori assumptions and can exhibit substantial modeling errors for multi-regime combustion characteristics. A computationally inexpensive, albeit non-generic, approach to remedy these shortcomings is the blending between premixed and non-premixed 1D flamelet solutions, e.g. based on some sort of modified flame index [START_REF] Domingo | Partially premixed flamelets in LES of nonpremixed turbulent combustion[END_REF][START_REF] Fiorina | Approximating the chemical structure of partially premixed and diffusion counterflow flames using FPI flamelet tabulation[END_REF][START_REF] Knudsen | A general flamelet transformation useful for distinguishing between premixed and non-premixed modes of combustion[END_REF][START_REF] Wen | LES of pulverized coal combustion with a multi-regime flamelet model[END_REF]. While this approach naturally recovers the limiting cases of premixed and non-premixed combustion, it ignores complex interactions between premixed and non-premixed reaction zones (i.e. multi-regime characteristics) which are essential for the accurate prediction of certain flame structures [START_REF] Masri | Partial premixing and stratification in turbulent flames[END_REF][START_REF] Hartl | Assessing the relative importance of flame regimes in Raman/Rayleigh line measurements of turbulent lifted flames[END_REF][START_REF] Butz | Local flame structure analysis in turbulent CH4/air flames with multi-regime characteristics[END_REF].

A first attempt to describe this sort of interaction in a two-dimensional composition space was published by Domingo et al. [START_REF] Domingo | Large-eddy simulation of a lifted methane jet flame in a vitiated coflow[END_REF] who derived flamelet-like equations with respect to the conditioning variables mixture fraction and progress variable. They showed that the modeling approach can distinguish between auto ignition, premixed, partially-premixed, and non-premixed combustion based on a series of Damköhler numbers. The approach was extended and generalized in the subsequent work by Nguyen et al. [START_REF] Nguyen | Multidimensional flamelet-generated manifolds for partially premixed combustion[END_REF] further including the numerical solution of 2D composition space equations. A slightly different approach was presented by Knudsen and Pitsch [START_REF] Knudsen | Capabilities and limitations of multi-regime flamelet combustion models[END_REF][START_REF] Knudsen | A general flamelet transformation useful for distinguishing between premixed and non-premixed modes of combustion[END_REF] who derived 2D composition space equations with respect to the mixture fraction and the reaction progress parameter Λ, a scalar quantity which is set to the value of progress variable at stoichiometry and held fixed across iso-mixture fraction surfaces. Using the progress-variable-like scalar Λ as a conditioning variable instead of the progress variable itself has the advantage that it is statistically independent from the mixture fraction. Nevertheless, other difficulties arise: it is unfeasible to transport Λ in a CFD context since its transport equation is too complex and it is not straight-forward to recover the premixed and non-premixed mode of combustion from the equations [START_REF] Knudsen | A general flamelet transformation useful for distinguishing between premixed and non-premixed modes of combustion[END_REF]. For the sake of completeness other two-dimensional composition space models are mentioned here: equation sets were published utilizing two mixture fractions as conditioning variables to model multiple fuel injections in Diesel engines [START_REF] Hasse | A two mixture fraction flamelet model applied to split injections in a DI diesel engine[END_REF][START_REF] Doran | A priori testing of a two-dimensional unsteady flamelet model for three-feed combustion systems[END_REF], and utilizing mixture fraction and enthalpy to capture mixing and thermal stratification in HCCI engines [START_REF] Cook | Numerical investigation of unburnt hydrocarbon emissions in a homogeneous-charge late-injection diesel-fueled engine[END_REF][START_REF] Mittal | An extended multi-regime flamelet model for IC engines[END_REF]. Similar approaches have also been presented for Conditional Moment Closure (CMC) [START_REF] Kronenburg | Double conditioning of reactive scalar transport equations in turbulent nonpremixed flames[END_REF].

Despite the more advanced representation of reaction zones, 2D composition space modeling has not been widely adopted, yet. This has several reasons: first, 2D composition space models rely on closure strategies for the scalar dissipation rates (or gradients) of the conditioning variables and the so-called cross-terms which additionally appear in the equations. Only recently it has been shown that a closure for the progress variable gradient (or its scalar dissipation rate) requires an additional equation to be solved [START_REF] Scholtissek | A self-contained progress variable space solution method for thermochemical variables and flame speed in freely-propagating premixed flamelets[END_REF][START_REF] Scholtissek | A self-contained composition space solution method for strained and curved premixed flamelets[END_REF], since no general model exists for this quantity. Second, while the classical flamelet equations are simple to solve and implement, 2D composition space models are significantly more complex. Third, the aspect of the statistical independence of the conditioning variables, as addressed by Knudsen and Pitsch [START_REF] Knudsen | Capabilities and limitations of multi-regime flamelet combustion models[END_REF][START_REF] Knudsen | A general flamelet transformation useful for distinguishing between premixed and non-premixed modes of combustion[END_REF], is not always rigorously incorporated in the composition space transformation.

The main motivation of this work is to advance the multi-dimensional composition space theory to overcome the aforementioned shortcomings. More specifically, the objective of this paper is twofold: (1) develop a consistent theory in a two-dimensional composition space to describe multi-regime combustion (c.f. Sec. 2) and ( 2) analyze and validate the approach in terms of a well-known laminar triple flame setup (c.f. Sec. 3). Thereby, this work shall provide a conceptual basis for modeling multi-regime combustion in CFD simulations with tabulated chemistry methods.

General formulation

In this section composition space equations for termochemical scalars are derived and discussed. The derivation consists of three steps: in Sec. 2.1, the evolution of thermochemical quantities is expressed in terms of mixture fraction isosurfaces serving as a Lagrangian frame of reference. Thereafter, an orthogonal coordinate system is introduced and used to project thermochemical scalars into composition space (Sec. 2.2). The composition space equations are presented and examined in Sec. 2.3.

It is then demonstrated how the composition space equations recover the limiting cases of combustion (Sec. 2.4), which is followed by a discussion of conceivable closures, cross-terms and differential diffusion (Sec. 2.5).

Lagrangian reference frame

Let us introduce the mixture fraction Z(x, t), a passive scalar (Z = 1 in fuel feeding stream and Z = 0 in oxidiser stream), whose balance equation reads (unity Schmidt and Lewis numbers) [START_REF] Bilger | The structure of diffusion flames[END_REF]

ρ ∂Z ∂t + ρu • ∇Z = ∇ • (ρD∇Z) , (1)
where ρ is the density, u is the velocity vector and D is the diffusion coefficient. The motion of an arbitrary Z-isosurface is described by the kinematic condition [START_REF] Matalon | On flame stretch[END_REF]

DZ Dt = ∂Z ∂t + u p • ∇Z = 0 , (2)
where u p can be interpreted as the velocity of a mass-less particle p attached to the Z-isosurface. Using Eq. (2) to replace the transient term in Eq. ( 1) yields [START_REF] Scholtissek | In-situ tracking of mixture fraction gradient trajectories and unsteady flamelet analysis in turbulent non-premixed combustion[END_REF]

u p = u - 1 ρ|∇Z| ∇ • (ρD∇Z) n , (3)
where the unit vector n = ∇Z/|∇Z| is the normalized mixture fraction gradient.

Let us now consider the chemical species Y i (x, t) (analogously other thermochemical scalars) verifying a balance equation of the usual form (unity Lewis numbers)

ρ ∂Y i ∂t + ρu • ∇Y i = ∇ • (ρD∇Y i ) + ωi , ( 4
)
where ωi is the chemical source. The evolution of the the species Y i at the particle position is defined by the material derivative

DY i Dt = ∂Y i ∂τ = ∂Y i ∂t + u p • ∇Y i , (5)
where τ is a time-like variable which is independent from the coordinate variables attached to the particle p (i.e. on a Z-isosurface). Using Eq. ( 4) to replace the transient term ∂Y i /∂t we can rewrite Eq. ( 5)

∂Y i ∂τ = 1 ρ ∇ • (ρD∇Y i ) + ωi ρ - 1 ρ ∇ • (ρD∇Z) ∂Y i ∂Z , (6)
where the partial derivative of Y i with respect to Z is given by

∂Y i ∂Z = 1 |∇Z| n • ∇Y i = ∇Z • ∇Y i |∇Z| 2 . ( 7
)
Equation 6 represents a species balance equation valid for any Z-isosurface and forms the basis for the following derivation. In case important physics occur along the normal n only, i.e. as for a thin non-premixed flame, a one-dimensional structure can be assumed and the composition space transformation of Eq. 6 leads to the unsteady flamelet equations [START_REF] Peters | Laminar flamelet concepts in turbulent combustion[END_REF]. For multi-regime combustion however, at least a second conditioning variable is required, which is considered next.

Orthogonal scalar coordinates

An orthogonal composition-space system of coordinates (n 1 , n 2 , n 3 ) may be constructed from Z and any reactive scalar Y c (x, t) described by

ρ ∂Y c ∂t + ρu • ∇Y c = ∇ • (ρD∇Y c ) + ωc , (8)
Figure 1: Sketch of the orthogonal scalar coordinate system in the physical space.

using the corresponding vector coordinates (Fig. 1)

n 1 = ∇Z |∇Z| , (9)
n 2 = ∇Y c -(n 1 • ∇Y c )n 1 |∇Y c -(n 1 • ∇Y c )n 1 | = ∇ ⊥ Y c |∇ ⊥ Y c | , (10)
n 3 = n 1 × n 2 = ∇e |∇e| . (11)
Here, the reactive scalar Y c is defined as a weighted sum of species

Y c = ns i α i Y i , (12)
a common definition in the literature [START_REF] Van Oijen | State-of-the-art in premixed combustion modeling using flamelet generated manifolds[END_REF][START_REF] Pierce | Progress-variable approach for large-eddy simulation of non-premixed turbulent combustion[END_REF][START_REF] Lodier | Composition-space premixed flamelet solution with differential diffusion for in situ flamelet-generated manifolds[END_REF]] and it will be denoted as reaction progress variable in the following. The gradient

∇ ⊥ Y c = ∇Y c -(n 1 • ∇Y c )n 1 (7) = ∇Y c - ∂Y c ∂Z ∇Z , (13)
is the orthogonal component of ∇Y c with respect to ∇Z and e is a coordinate along of which changes of thermochemical quantities are assumed small (similar as in [START_REF] Knudsen | A general flamelet transformation useful for distinguishing between premixed and non-premixed modes of combustion[END_REF]). The definition of the second coordinate, Eq. (10), ensures orthogonality, i.e. n 1 • n 2 = 0. In this (n 1 , n 2 , n 3 )

Figure 2: In the physical space (left), two mixture fraction gradient trajectories (GT) and two Z-isolines intersect at four points. The same points are mapped into the composition space parametrized by Z and Y c (right). Traversing points along the gradient trajectories and Z-isolines in the physical space leads to a corresponding motion in composition space in the direction of the red and blue arrows, respectively.

orthogonal coordinate system, usual projection rules apply and ∇Y i reads

∇Y i = (n 1 • ∇Y i ) n 1 + (n 2 • ∇Y i ) n 2 + (n 3 • ∇Y i ) n 3 , (14)
= ∂Y i ∂Z |∇Z|n 1 + ∂Y i ∂Y c Z |∇ ⊥ Y c |n 2 + ∂Y i ∂e |∇e|n 3 (15)
= ∂Y i ∂Z ∇Z + ∂Y i ∂Y c Z ∇ ⊥ Y c + ∂Y i ∂e ∇e . ( 16
)
where the definition

∂Y i ∂Y c Z = ∇Y i • ∇ ⊥ Y c |∇ ⊥ Y c | 2 , (17)
is used to represent the derivative of the thermochemical quantity Y i projected into composition space along ∇ ⊥ Y c . Note that this derivative of Y i with respect to Y c is defined for constant mixture fractions, which is indicated by the notation (•)| Z and directly follows from the construction of the orthogonal coordinate system (c.f. Fig. 1). As previously stated, derivatives of thermochemical variables with respect to e are assumed small and are therefore omitted in the following. Before further usage, it is useful to examine the mapping from physical to composition space by the orthogonal coordinates. Although Z and Y c are generally non-orthogonal, these coordinates are often used to parametrize the composition space for reaction zone analyses and we also adopt this representation here. Figure 2 schematically shows intersections of two exemplary mixture fraction gradient trajectories (GT) and two Z-isolines in a two-dimensional plane. It is assumed that the shaded region is located in the vicinity of a reaction zone, i.e. an area with significant chemical activity and heat release. Traversing points along a gradient trajectory in the physical space (i.e. constructing a trajectory following ∇Z), both mixture fraction and progress variable change, which results in a corresponding motion in composition space (red arrow). In case there are no progress variable gradients along Z-isolines (i.e. ∇ ⊥ Y c = 0) this motion is analogous to the traversal of a non-premixed flame. On the contrary, traversing points along Z-isolines (i.e. along ∇ ⊥ Y c ) results in a motion in composition space along Y c only (blue arrow), while Z remains constant. If mixture gradients are sufficiently small this motion corresponds to the traversal of a one-dimensional premixed flame (neglecting differential diffusion).

Composition space equations for species and temperature

Introducing Eq. 16 1 into Eq. 6 and multiplying by the density yields

ρ ∂Y i ∂τ = ρD∇Z • ∇ ∂Y i ∂Z + ρD∇ ⊥ Y c • ∇ ∂Y i ∂Y c Z + ∇ • (ρD∇ ⊥ Y c ) ∂Y i ∂Y c Z + ωi , = ρD|∇Z| 2 ∂ 2 Y i ∂Z 2 + ρD|∇ ⊥ Y c | 2 ∂ 2 Y i ∂Y c 2 Z + ∇ • (ρD∇ ⊥ Y c ) ∂Y i ∂Y c Z + ωi , (18)
In the expression above, Eq. ( 15) was used to reformulate the first and second term on the right hand side according to

ρD∇Z • ∇ ∂Y i ∂Z = ρD|∇Z| (n 1 • n 1 ) |∇Z| ∂ ∂Z ∂Y i ∂Z + ρD|∇Z| (n 1 • n 2 ) |∇ ⊥ Y c | ∂ ∂Y c Z ∂Y i ∂Z = ρD|∇Z| 2 ∂ 2 Y i ∂Z 2 , (19)
where

n 1 • n 1 = 1 and n 1 • n 2 = 0 (orthogonality) was used.
With the intention to replace the third term on the right hand side of Eq. ( 18), a summation analogous to Eq. ( 12) is applied to Eq. ( 18)

ρ ∂Y c ∂τ = ρD|∇Z| 2 ∂ 2 Y c ∂Z 2 + ρD|∇ ⊥ Y c | 2 ∂ 2 Y c ∂Y c 2 Z =0 +∇ • (ρD∇ ⊥ Y c ) ∂Y c ∂Y c Z =1 + ωc , (20)
Thus, Eq. ( 20) simplifies to

ρ ∂Y c ∂τ = ∇ • (ρD∇ ⊥ Y c ) + ωc ⊥ , (21)
with the source

ωc ⊥ = ρD|∇Z| 2 ∂ 2 Y c ∂Z 2 + ωc . (22)
Equation ( 21) balances the progress variable Y c on Z-isosurfaces. Therefore, transport along the direction of Z-gradients (i.e. between different Z-isosurfaces) is attributed to the source term in Eq. ( 22). This term can also be interpreted as a correction term which ensures orthogonality between ∇Z and the newly introduced ∇ ⊥ Y c . From another perspective, the above expression for the source term is equivalent to the equation for an unsteady, non-premixed laminar flamelet [START_REF] Peters | Laminar flamelet concepts in turbulent combustion[END_REF] and in this context ωc ⊥ corresponds to the transient term.

This already indicates a suitable mapping between time and reaction progress variable, which has already been utilized for unsteady flamelet modeling [START_REF] Ihme | Prediction of autoignition in a lifted methane/air flame using an unsteady flamelet/progress variable model[END_REF] and modeling of homogeneous reacting mixtures [START_REF] Domingo | Large-eddy simulation of a lifted methane jet flame in a vitiated coflow[END_REF][START_REF] Scholtissek | A self-contained composition space solution method for strained and curved premixed flamelets[END_REF] in previous works. This aspect is revisited and discussed further below.

In a CFD simulation, it would be impractical to solve Eq. ( 21) since this would require Z-isosurfaces to be tracked in time and space. However, the equation is used to replace the third term on the right hand side of Eq. ( 18) which yields

ρ ∂Y i ∂τ = ρD|∇Z| 2 ∂ 2 Y i ∂Z 2 + ρD|∇ ⊥ Y c | 2 ∂ 2 Y i ∂Y c 2 Z + ρ ∂Y c ∂τ ∂Y i ∂Y c Z -ωc ⊥ ∂Y i ∂Y c Z + ωi . ( 23
)
This equation is simplified assuming that the thermochemical state can be approximated by a reduced order manifold, which is parametrized by mixture fraction and progress variable. Then, a first order taylor series expansion of the first term in Eq. ( 23) can be written as

∂Y i ∂τ ≈ ∂Z ∂τ =0 ∂Y i ∂Z + ∂Y c ∂τ ∂Y i ∂Y c Z = ∂Y c ∂τ ∂Y i ∂Y c Z , (24)
which states that the transient evolution of the thermochemical state at the particle position (i.e. at constant mixture fraction) is related to the transient evolution of the progress variable at this point. Introducing the expression into Eq. ( 23) the transient terms compensate each other and vanish. It has been shown in previous works [START_REF] Scholtissek | A self-contained progress variable space solution method for thermochemical variables and flame speed in freely-propagating premixed flamelets[END_REF][START_REF] Scholtissek | A self-contained composition space solution method for strained and curved premixed flamelets[END_REF] that composition space models can accurately recover characteristics of transient combustion processes (e.g. auto-ignition, spherical expanding flames) based on this approximation. This also applies to complex transient processes such as two-stage ignition [START_REF] Scholtissek | A self-contained composition space solution method for strained and curved premixed flamelets[END_REF]. Equation ( 24) is based on the fact that chemical reaction mechanisms propagate the change in one species through a network of chemical reactions to many others. Thereby, only few species, which are directly or indirectly affected by the most relevant reaction pathways, are required in the definition of Y c (c.f. Eq. ( 12)). Nevertheless, it is noted that Eq. ( 24) can lead to modeling errors for very large transients experienced by certain intermediate species, such as radicals.

Introducing Eq. ( 24) into Eq. ( 23) and performing an analogous transformation for the temperature, which is subject to the transport equation (unity Lewis numbers)

ρ ∂T ∂t + ρu • ∇T = 1 c p ∇ • (λ∇T ) + ns k c p,k c p ρD∇Y k • ∇T + ωT , (25)
yields the two-dimensional composition space equations

ωc ⊥ ∂T ∂Y c Z = ρD|∇Z| 2 ∂ 2 T ∂Z 2 + ρD|∇ ⊥ Y c | 2 ∂ 2 T ∂Y c 2 Z + ρD c p |∇Z| 2 ∂c p ∂Z ∂T ∂Z + |∇ ⊥ Y c | 2 ∂c p ∂Y c Z ∂T ∂Y c Z + ρD ns k c p,k c p |∇Z| 2 ∂Y k ∂Z ∂T ∂Z + |∇ ⊥ Y c | 2 ∂Y k ∂Y c Z ∂T ∂Y c Z + ωT , (26)
ωc ⊥ ∂Y i ∂Y c Z Yc-drift = ρD|∇Z| 2 ∂ 2 Y i ∂Z 2 Z-diffusion + ρD|∇ ⊥ Y c | 2 ∂ 2 Y i ∂Y c 2 Z Yc-diffusion + ωi source . (27)
Considering the species equation above, it consists of a drift term, two diffusive transport terms along the directions of Z and (orthogonal) Y c , respectively, and a chemical source term. The drift term, formally a convective term in composition space, incorporates the evolution of the reaction progress variable itself. An equivalent term does not exist for the mixture fraction since it is a passive scalar. Opposed to similar previous works [START_REF] Knudsen | A general flamelet transformation useful for distinguishing between premixed and non-premixed modes of combustion[END_REF][START_REF] Nguyen | Multidimensional flamelet-generated manifolds for partially premixed combustion[END_REF][START_REF] Hasse | Modelling of ignition mechanisms and pollutant formation in direct-injection diesel engines with multiple injections[END_REF], Eqs. ( 26) and ( 27) do not contain so-called cross-terms due to orthogonality of the employed coordinate system. Comparing the equation set to the one presented by Nguyen et al. [START_REF] Nguyen | Multidimensional flamelet-generated manifolds for partially premixed combustion[END_REF] (c.f. Eqs. ( 21) and ( 22) of their work), who conditioned temperature and species on (Z, Y c ) without explicitly incorporating orthogonality, the two main differences besides the removal of cross-terms are: (1) the correction for the progress variable source term ωc ⊥ (c.f. Eq. ( 22)), and (2) the modified gradient |∇ ⊥ Y c | scaling diffusion along the Y c -dimension (keeping Z fixed). The absence of cross-terms is examined further in Sec. 2.5. Before, it is demonstrated that the equations recover the asymptotic limits of combustion.

Recovery of limiting cases of combustion

The composition space equations describing (i) a homogeneous reacting mixture, (ii) a premixed flamelet, and (iii) a non-premixed flamelet can be readily obtained as a subset of Eqs. ( 26) and [START_REF] Kronenburg | Double conditioning of reactive scalar transport equations in turbulent nonpremixed flames[END_REF].

Homogeneous reacting mixture. In the absence of spatial gradients (|∇Z| = |∇ ⊥ Y c | = 0), the mixture fraction is uniform (Z = const.). In this case, the balance equation for Y c on Z-isosurfaces, Eq. 21, becomes equivalent to the Y c -equation, Eq. ( 8), and the homogeneous reactive mixture only evolves in time. Equations ( 26) and ( 27) simplify to ωc ∂T ∂Y c = ωT and ωc

∂Y i ∂Y c = ωi . ( 28
)
It has been shown [START_REF] Domingo | Large-eddy simulation of a lifted methane jet flame in a vitiated coflow[END_REF][START_REF] Scholtissek | A self-contained composition space solution method for strained and curved premixed flamelets[END_REF] that these equations accurately describe the evolution of a constant pressure homogeneous reacting mixture (ignition problem) in composition space. Note that a time-like coordinate in composition space is not required since the unsteadiness of thermochemical variables in the physical space is inherently captured through the transient evolution of the conditioning variables (here only of Y c ), c.f. Eq. ( 24).

Premixed flamelet. In the limit of a premixed flame (assuming Le = 1), the mixture fraction gradient is zero, ∇Z = 0. From Eq. ( 13) it is obvious that ∇ ⊥ Y c = ∇Y c . Furthermore, the correction term vanishes in Eq. ( 22) and ωc ⊥ = ωc . In composition space, the Eqs. ( 26) and ( 27) consistently simplify to the 1D premixed flamelet equations [START_REF] Scholtissek | A self-contained progress variable space solution method for thermochemical variables and flame speed in freely-propagating premixed flamelets[END_REF][START_REF] Lodier | Composition-space premixed flamelet solution with differential diffusion for in situ flamelet-generated manifolds[END_REF][START_REF] Savard | Effects of dissipation rate and diffusion rate of the progress variable on local fuel burning rate in premixed turbulent flames[END_REF]:

ωc ∂T ∂Y c = ρD|∇Y c | 2 ∂ 2 T ∂Y c 2 + ρD c p |∇Y c | 2 ∂c p ∂Y c ∂T ∂Y c + ρD ns k c p,k c p |∇Y c | 2 ∂Y k ∂Y c ∂T ∂Y c + ωT , (29)
ωc ∂Y i ∂Y c = ρD|∇Y c | 2 ∂ 2 Y i ∂Y c 2 + ωi . (30)
Non-premixed flamelet. For a purely non-premixed flame, Y c can be expressed as a function of Z which implies that gradients ∇Z and ∇Y c are aligned. It follows for Eq. ( 13)

∇ ⊥ Y c = ∇Y c - ∂Y c ∂Z ∇Z = 0 . ( 31
)
Equation 16 then reduces to the conventional flamelet transformation rule ∇T = ∇Z ∂T ∂Z and the balance equation for Y c on Z-isosurfaces, Eq. 21, simplifies to

ρ ∂Y c ∂τ = ωc ⊥ = ρD|∇Z| 2 ∂ 2 Y c ∂Z 2 + ωc . (32)
Accordingly, one obtains from Eqs. ( 26) and ( 27)

ωc ⊥ ∂T ∂Y c Z = ρD|∇Z| 2 ∂ 2 T ∂Z 2 + ρD c p |∇Z| 2 ∂c p ∂Z ∂T ∂Z + ρD ns k c p,k c p |∇Z| 2 ∂Y k ∂Z ∂T ∂Z + ωT , (33)
ωc ⊥ ∂Y i ∂Y c Z = ρD|∇Z| 2 ∂ 2 Y i ∂Z 2 + ωi . ( 34
)
It is readily shown that the above equation set is equivalent to the unsteady 1D flamelet equations for non-premixed combustion [START_REF] Peters | Laminar flamelet concepts in turbulent combustion[END_REF] by replacing the drift term on the left hand side according to

ωc ⊥ ∂Y k ∂Y c Z (32) = ρ ∂Y c ∂τ ∂Y k ∂Y c Z (24) = ρ ∂Y k ∂τ . ( 35
)
This illustrates again the relationship between time and the reaction progress variable. Note that if only the mixture fraction Z was chosen as a conditioning variable (without a suitable coordinate capturing reaction progress), the transient term remains in the equations during the derivation presented in Sec. 2.3.

Discussion of conceivable closures, cross-terms and differential diffusion

The composition space theory presented so far can be used for analyses of fully resolved numerical datasets (c.f. Sec. 3). Further, an algorithm for the numerical solution of Eqs. ( 26) and ( 27) would be very valuable for modeling multi-regime combustion. A reduced order manifold could thereby be constructed directly in composition space and tabulated as a function of a reduced set of scalars (mixture fraction, progress variable and others). While such an algorithm is subject to future research, three key issues for this effort are briefly discussed here: (i) closures for the gradients |∇Z| and |∇ ⊥ Y c |, (ii) cross-terms, and (iii) an extension to capture differential diffusion (non-unity Lewis numbers).

As stated above, a numerical solution algorithm for Eqs. ( 26) and ( 27) requires closures for |∇Z| and |∇ ⊥ Y c |. From the analytical solution of mixing layers [START_REF] Peters | Laminar diffusion flamelet models in non-premixed turbulent combustion[END_REF] promising closures exist for |∇Z| (or the scalar dissipation rate χ Z = 2D|∇Z| 2 ). On the contrary, |∇ ⊥ Y c | depends on the progress variable gradient |∇Y c | which substantially changes with the fuel-oxidizer combination or the specific progress variable definition (c.f. Eq. ( 12)) [START_REF] Scholtissek | A self-contained progress variable space solution method for thermochemical variables and flame speed in freely-propagating premixed flamelets[END_REF] and no generic analytical closure exists. In our previous works [START_REF] Scholtissek | A self-contained progress variable space solution method for thermochemical variables and flame speed in freely-propagating premixed flamelets[END_REF][START_REF] Scholtissek | A self-contained composition space solution method for strained and curved premixed flamelets[END_REF], a separate equation for the progress variable gradient has been utilized and it is likely that this approach can be extended to complement the composition space equations presented here. This aspect should be explored in future work.

Further, comparing Eqs. ( 26) and ( 27) to the equations derived by Nguyen et al. [START_REF] Nguyen | Multidimensional flamelet-generated manifolds for partially premixed combustion[END_REF], it is apparent that thermochemical quantities are evolving in four dimensions here,

i.e. Y i (Z, Y c | Z , |∇Z|, |∇ ⊥ Y c |), opposed to five dimensions in [22], i.e. Y i (Z, Y c , χ Z , χ Yc , χ Z,Yc ). While |∇Z| is readily related to the scalar dissipation rate χ Z = 2D|∇Z| 2 , |∇ ⊥ Y c | is linked to both χ Yc = 2D|∇Y c | 2
and the cross-scalar dissipation rate χ Z,Yc = 2D∇Z • ∇Y c . Without orthogonal coordinates, cross-terms appear in composition space equations [START_REF] Knudsen | A general flamelet transformation useful for distinguishing between premixed and non-premixed modes of combustion[END_REF][START_REF] Nguyen | Multidimensional flamelet-generated manifolds for partially premixed combustion[END_REF][START_REF] Hasse | Modelling of ignition mechanisms and pollutant formation in direct-injection diesel engines with multiple injections[END_REF] with χ Z,Yc as a scaling factor. Cross-terms require a closure and are difficult to handle from a modeling perspective. So far no general closure strategy has been presented and sometimes these terms are neglected accepting a certain modeling error. The cross-scalar dissipation rate χ Z,Yc is defined based on the scalar product

∇Z • ∇Y c = χ Z,Yc 2D = |∇Z| 2 ∂Y c ∂Z = |∇Z| |∇Y c | cos ϕ , ( 36
)
where ϕ is the enclosed angle between the gradients ∇Z and ∇Y c . Furthermore,

|∇ ⊥ Y c | is related to |∇Y c | through ϕ |∇ ⊥ Y c | 2 = ∇ ⊥ Y c • ∇ ⊥ Y c = ∇ ⊥ Y c • ∇Y c -∇Z ∂Y c ∂Z = ∇ ⊥ Y c • ∇Y c = |∇Y c | 2 -∇Z • ∇Y c ∂Y c ∂Z = |∇Y c | 2 (1 -cos 2 ϕ) , (37)
which is linked to the definition of the second coordinate n 2 of the orthogonal coordinate system introduced with Eq. [START_REF] Fiorina | Approximating the chemical structure of partially premixed and diffusion counterflow flames using FPI flamelet tabulation[END_REF]. Hence, the absence of cross-terms in Eqs. ( 26) and ( 27) is achieved by incorporating the interdependence between progress variable and mixture fraction into the construction of the orthogonal coordinates.

As previously stated, Z and Y c are generally non-orthogonal coordinates. This affects the numerical solution of Eqs. ( 26) and ( 27): if the composition space equations ( 26) and ( 27) are discretized on a (Z, Y c )-grid, which is for that purpose made orthogonal, partial derivatives with respect to Z have to be split into a component keeping Y c constant and cross-terms are re-introduced as correction terms to account for the non-orthogonality of (Z, Y c ). This issue remains beyond the scope of this paper, but it should be noted that the numerical solution of the composition space equations is therefore not trivial.

Nevertheless, the combination of Z and Y c as conditioning variables for thermochemical scalars is a natural choice for combustion processes and these quantities should be kept to parametrize numerical solutions of Eqs. ( 26) and [START_REF] Kronenburg | Double conditioning of reactive scalar transport equations in turbulent nonpremixed flames[END_REF]. In the context of tabulated chemistry approaches, already existing turbulence-chemistry interaction models, as required for Large-Eddy Simulations, can be used with minor modifications. Even if the composition space equations are eventually solved in a different reference frame, mixture fraction and progress variable can be reconstructed for tabulation and look-up.

A worthwhile extension of the present composition space theory is the consistent represen-tation of differential diffusion (non-unity Lewis numbers). Previous works on one-dimensional composition space modeling for premixed [START_REF] Scholtissek | A self-contained progress variable space solution method for thermochemical variables and flame speed in freely-propagating premixed flamelets[END_REF][START_REF] Scholtissek | A self-contained composition space solution method for strained and curved premixed flamelets[END_REF][START_REF] Lodier | Composition-space premixed flamelet solution with differential diffusion for in situ flamelet-generated manifolds[END_REF] and non-premixed combustion [START_REF] Pitsch | Unsteady flamelet modeling of turbulent hydrogen-air diffusion flames[END_REF][START_REF] Dietzsch | The impact of thermal diffusion on the structure of non-premixed flames[END_REF] have shown that this can be realized conditioning on either mixture fraction or progress variable. The extension of the two-dimensional composition space equations for differential diffusion requires a careful incorporation of detailed diffusion modeling leading to additional terms and transport effects, such as curvature-induced differential diffusion [START_REF] Scholtissek | Flamelet budget and regime analysis for non-premixed tubular flames[END_REF]. This extension for the Eqs. ( 26) and ( 27) is therefore non-trivial, but highly important for describing multi-regime combustion without restrictive assumptions regarding diffusive transport.

Analysis of a laminar triple flame

The composition space equations presented in the previous section are validated with the fully resolved, detailed chemistry solution of a laminar methane-air triple flame. Figure 3 shows a schematic of the triple flame setup. It consists of a lean and a rich premixed flame branch which enclose a non-premixed flame. The three different flame structures merge at the characteristic triple point, which plays an important role for the flame stabilization. The triple flame is established by defining an inflow of premixed fresh gases (T 0 = 300 K, p = 1 atm) with a mixture stratification in the cross-flow direction. The stoichiometric mixture fraction is Z st = 0.055 (methane-air), and the boundary conditions are chosen as Z min = 0 (pure air) and Z max = 0.42. The boundary conditions in the cross-flow direction are implemented as slip walls. The imposed mixture fraction gradient at the inlet is 50 m -1 which results in an edge-flame aspect ratio of f = 0.024. The latter expresses a ratio between a diffusion length measured along the stoichiometric surface (premixed kernel) and a diffusion length across the stoichiometric surface (diffusion flame) [START_REF] Boulanger | Diffusion edge-flame: approximation of the flame tip Damköhler number[END_REF]. With f = 0.024, the laminar triple flame corresponds to a weakly curved partially premixed flame according the classification introduced in [START_REF] Boulanger | Diffusion edge-flame: approximation of the flame tip Damköhler number[END_REF]. The GRI-MECH 3.0 reaction mechanism [START_REF] Smith | Gri-mech 3.0[END_REF] is used for describing chemical kinetics and unity Lewis numbers are assumed for all species. Further information on the numerical setup of the triple flame can be found in [START_REF] Hartl | Regime identification from Raman/Rayleigh line measurements in partially premixed flames[END_REF]. From the fully resolved scalar fields of this flame all terms which appear in the 2D composition space equations, Eqs. ( 26) and ( 27), are reconstructed and evaluated.

Triple flame characteristics

Simulation results obtained for the scalar fields of temperature, heat release rate (HRR) and four species mass fractions of the laminar triple flame are displayed in Fig. 4. The largest heat release rate is found in the vicinity of the triple point, where the conversion of reactants (c.f. O 2 ) to hot products (c.f. CO 2 ) and the maximum heat release occur in a thin reactive layer. Further, radical species, such as OH, are found to reach a maximum there. The thickness of the reactive layer, which can be assessed from the transition region of reactants to products, increases downstream of the triple point marking the two premixed flame branches. The upper rich premixed flame appears stronger than the lower lean premixed flame since its contribution to the overall heat release is larger. The heat release by the non-premixed flame remains at least one order of magnitude smaller than that of the premixed flames which is why it remains hardly visible in the HRR plot. As expected, considerably more CO is formed in the rich premixed flame compared to the lean side of the triple flame. The overall maximum of temperature and reaction products (such as CO 2 ) is eventually reached at the position of the non-premixed flame, where conditions approach the chemical equilibrium. Similar characteristics are found from Fig. 6, which shows the angle between ∇Z and ∇Y c (left) and gradient trajectories of ∇Z, ∇Y c , and ∇ ⊥ Y c , respectively (middle and right). Note that the definition of the angle is slightly modified here

ϕ * = arccos ∇Z • ∇Y c |∇Z| |∇Y c | , (38)
such that ϕ * = 0 • indicates a full alignment between ∇Z and ∇Y c and ϕ * = 90 • indicates orthogonality. From both, Figs. 6 (left) and 6 (middle), it is apparent that ∇Z and ∇Y c are strongly misaligned (ϕ * = 50 . . . 90 • ) upstream from and also directly at the position of the triple point. On the other hand, ∇Z and ∇Y c become almost perfectly aligned (ϕ * ≈ 0 • ) for the downstream region x > 5 mm. These characteristics suggest that Z-Y c cross-terms, corresponding to non-negligible values for 0 • < ϕ * < 90 • and significant gradients of |∇Z| and |∇Y c |, play an important role for the stabilization mechanism and the most reactive region of the triple flame. According to the considerations in Sec. 2.4, these characteristics further imply multi-regime combustion due to deviations from the premixed (negligible |∇Z|) and the non-premixed (ϕ * ≈ 0 • ) burning mode. On the other hand, gradient alignment in the downstream region (x > 5 mm) suggests that only one conditioning variable is necessary to describe local reaction zones as either the progress variable can be expressed as a function of the mixture fraction (non-premixed limit) or the mixture fraction gradient tends to zero (premixed limit). Additionally, the right plot displays gradient trajectories for ∇Z and ∇ ⊥ Y c confirming that both gradients are perpendicular throughout the computational domain.

Gradient-free regime analysis

The gradient-free regime identification (GFRI) approach [START_REF] Hartl | Assessing the relative importance of flame regimes in Raman/Rayleigh line measurements of turbulent lifted flames[END_REF][START_REF] Hartl | Regime identification from Raman/Rayleigh line measurements in partially premixed flames[END_REF] allows the identification of the local combustion regime based on experimental and numerical data. Starting from the fully resolved numerical solution of the triple flame, the mixture fraction Z, the heat release rate (HRR), and the chemical mode (CM) are calculated and used to detect and characterize reaction zones. It was shown that strong correlations between the magnitude of change in the CM zero-crossing, the mixture fraction at the CM zero-crossing, and the local HRR at the CM zero-crossing exist [START_REF] Hartl | Assessing the relative importance of flame regimes in Raman/Rayleigh line measurements of turbulent lifted flames[END_REF]. Hence, the combination of Z, HRR, and CM allows the identification of dominantly premixed, dominantly non-premixed, and intermediate structures in which both premixed and non-premixed reaction zones contribute to the overall heat release rate. Further, the approach was extended and an automated classification of local reaction zone structures in complex multi-regime flames was derived [START_REF] Butz | Local flame structure analysis in turbulent CH4/air flames with multi-regime characteristics[END_REF]. Based on the ratio of local HRR peaks associated with premixed (HRR P ) and non-premixed (HRR NP ) regions located in close spatial proximity, a classification into premixed, dominantly premixed, multi-regime, dominantly non-premixed, and non-premixed flame zones is possible by using the flame structure index η:

η = HRR P -HRR NP HRR max , (39)
where HRR max is the HRR maximum in close spatial proximity of the local reaction zones. The criteria for the classification of reaction zones are summarized in Table 1.

Table 1: Criteria for the classification of reaction zones. Figure 7 shows the distribution of the aforementioned combustion regimes for the laminar triple flame as a contour plot (top). From this figure the complex flame structure consisting of lean and rich premixed flame branches with an embedded non-premixed flame is visible. It is found that the intense HRR in the vicinity of the triple point is mainly driven by premixed, and multi-regime regions. Premixed and multi-regime structures exists until an axial position of x = 4 mm and are followed by dominantly premixed flame branches. These dominantly premixed flame zones are characterized by a maximum HRR which is shifted to the position of the CM zero-crossing, an indicator for slightly weakened premixed flames.

Starting from x = 3 mm, additionally non-premixed characteristics arise along the flame centerline until the end of the computational domain. It is emphasized that non-premixed structures also exist between the triple point and x = 3 mm, however, the elevated local HRR values cannot be explicitly attributed to one specific flame zone due to the strong interaction between premixed, multi-regime and non-premixed zones. The lower premixed flame branch is characterized by the premixed and dominantly premixed flame region and the upper premixed flame branch by multi-regime and dominantly-premixed flame zones. Note, that multi-regime structures can only exist in rich to slightly rich regions, allowing premixed and non-premixed flame zones to exist in close proximity.

For reference, the normalized flame index is shown at the bottom of Fig. 7. It is evaluated from [START_REF] Fiorina | Approximating the chemical structure of partially premixed and diffusion counterflow flames using FPI flamelet tabulation[END_REF]

ξ = ∇Y fuel • ∇Y oxid |∇Y fuel • ∇Y oxid | , (40)
where Y fuel = Y CH 4 and Y oxid = Y O 2 denote the fuel and oxidizer mass fractions, respectively. From the flame index, premixed flames are identified by ξ = 1 and non-premixed flames by ξ = -1. Overall, it agrees well with the results obtained from the GFRI approach, although it is evident that the latter yields more refined information on the individual flame zones.

The findings from the extended GFRI approach are in agreement with the analysis of the triple flame characteristics above and existing flame zone analyses from literature [START_REF] Knudsen | A general flamelet transformation useful for distinguishing between premixed and non-premixed modes of combustion[END_REF][START_REF] Wu | A new multi-dimensional flamelet generated manifolds approach for approximating partially premixed flame structure[END_REF]. It provides a more detailed understanding of the complex flame structures close to the triple point and serves as an orientation for the budget analysis of the 2D composition space equations in the following.

Budget analysis of the 2D composition space equations

A detailed analysis of the single terms of Eq. ( 27) for the species CO is performed. CO is chosen since it has been challenging to model this intermediate species in the context of tabulated chemistry approaches [START_REF] Butz | Local flame structure analysis in turbulent CH4/air flames with multi-regime characteristics[END_REF]. All terms which appear in the two-dimensional species equation are computed from the solution of the laminar triple flame using detailed chemistry. Such an analysis allows to quantify: (1) the local importance of the individual terms, and (2) confirms that all relevant transport effects are considered in the equations, i.e. that the transport terms balance the local chemical source term. It will be shown further below that this is not the case for conventional flamelet models relying on the assumption of a 1D flame structure. Figure 8 shows all terms of the species equation for CO along five different vertical slices in the vicinity of the triple point. For a perfectly balanced equation the sum of these terms is zero, which can be inspected in Fig. 8 by the term balance.

Slightly upstream of the triple point ( 1 ), the chemical source term for CO is small and the dominant Z-diffusion (influence of stratification) is balanced by the Y c -drift term. Closer to the triple point [START_REF] Masri | Partial premixing and stratification in turbulent flames[END_REF], this characteristic prevails while the chemical source term and the Y c -transport term both increase. Directly at the triple point ( 3 ), the region with the highest chemical reactivity, the chemical source term becomes large and is compensated by all other terms, with the largest contribution from the Y c -transport term. This observation agrees with 27), for Y CO along five x-slices (vertical red lines in contour plot). All terms which appear in the 2D composition space equations were post-processed from the detailed chemistry solution of the triple flame.

the combustion regime identification which shows premixed and multi-regime characteristics close to the triple point. Further downstream ( 4 and 5 ), two peaks of the chemical source terms form, which are related to the premixed flame branches. The chemical source term for the rich premixed flame is larger which reconfirms that this flame branch is stronger compared to the lean premixed flame. Notably, in the budgets of the composition space equations for slices 4 and 5 all terms are relevant and non-negligible. With this, Fig. 8 suggests that none of the terms in the equations is negligible for the triple flame and that the premixed flame branches downstream of the triple point are also significantly affected by stratification effects. Figure 9 shows budgets of the 2D composition space equation for CO along horizontal slices of the triple flame. Five slices are extracted along the y-direction from upstream to downstream of the triple point. Slice 1 passes the lean premixed flame branch. It is found that at first the Z-diffusion and the Y c -drift term balance each other upstream of the flame. Moving along the x-direction a positive peak in the chemical source term (CO-production region) is found and at that position additionally the Y c -diffusion term gains relevance. The chemical source term then changes sign (CO-oxidation region) where the Y c -diffusion becomes negligible small. Throughout this whole flame structure Z-diffusion and Y c -drift are important. Considering the slices 2 and 3 , which are located closer to the triple point, similar flame structures can be identified. Nevertheless, the structure becomes notably thinner, moves further upstream and the relative influence of Z-diffusion declines, while 20 27), for Y CO along several y-slices (horizontal red lines in contour plot). All terms which appear in the 2D composition space equations were post-processed from the detailed chemistry solution of the triple flame.

Y c -diffusion becomes a dominant term. Hence, according to the budgets the flame exhibits distinct premixed characteristics, which is in accordance with the previous findings. Above the triple point ( 4 and 5 ), the Z-diffusion becomes more relevant again while Y c -diffusion declines. Due to the rich conditions in the upper premixed flame branch (c.f. 5 ) the chemical source term remains positive and there is no further CO-oxidation to CO 2 .

Both, Figs. 8 and9 show that the budgets of the 2D composition space equations closely reflect the regime characteristics also found from the GFRI analysis in Sec. 3.2. It becomes evident that different complex burning modes co-exist in the vicinity of the triple point which cannot solely be attributed to either premixed or non-premixed burning mode. Along these lines, Fig. 10 shows a comparison of the balance error of the 1D non-premixed flamelet equations (Eq. ( 34)), the 1D premixed flamelet equations (Eq. ( 30)) and the 2D composition space equations for multi-regime combustion (Eq. ( 27)). The equation balances are obtained from the summation of all terms which are post-processed from the detailed solution of the triple flame. Non-zero values indicate that the transport terms contained in the respective model do not fully balance the chemical source term. Note that the equation balances are normalized with the absolute value of the local chemical source term and they are only shown for points where chemical activity can be detected, i.e. where the source term exceeds 0.1 % of its global maximum value. The left column in Fig. 10 shows that the non-premixed flamelet equations (Z-flamelet) cannot capture the characteristics of the lean and rich premixed flame branches. On the other hand, the premixed flamelet equations (Y c -flamelet) in the middle column show substantial balance errors for H 2 O, CO and OH downstream of the triple point. As indicated by the color scheme, these balance errors can become of the same order of magnitude as the chemical source term. While the conventional 1D flamelet models show large balance errors in certain regions, the 2D composition space equations are accurate everywhere throughout the triple flame. Hence, both asymptotic modes of burning as well as their complex interactions are recovered by the 2D composition space model.

Summary and conclusions

In this work, a new set of two-dimensional composition space equations for modeling multi-regime combustion is derived. Thermochemical variables are conditioned on the mixture fraction Z and the reaction progress variable Y c . While the orientation of the first coordinate is defined by the mixture fraction gradient, the second coordinate direction is based on the gradient component of the progress variable that is perpendicular to ∇Z. This way, the interdependence between mixture fraction and progress variable is accounted for. The orthogonal set of coordinates is then used in the transformation of species and temperature equations to the two-dimensional composition space. Given certain simplifications, it is shown that the equations naturally reduce to conventional one-dimensional composition space equations for premixed and non-premixed combustion, and a homogeneous reacting mixture. Previous works have used non-orthogonal coordinates which lead to so-called cross-terms in the corresponding multi-dimensional composition space equations. These cross-terms require a closure, but to the authors' knowledge no general closure has been presented yet, if it exists at all. When using the orthogonal coordinates, cross-terms formally disappear.

In the second part of the paper, the two-dimensional composition space equations are validated with the detailed chemistry solution of a laminar methane-air triple flame. Postprocessing of this dataset allows the evaluation and examination of all terms appearing in the composition space equations. Together with a suitable combustion regime identification, it is shown that the novel equations accurately recover the balance of temperature and species everywhere in the triple flame. While conventional one-dimensional composition space equations fail particularly in the vicinity of the triple point, where multi-regime characteristics are detected, the novel two-dimensional equation set balances the chemical source term accurately. Moreover, the misalignment between mixture fraction and progress variable gradients suggest that Z-Y c cross-terms play a role in the stabilization mechanism and the flame structure around the triple point of the flame. This indicates that cross-terms are most likely non-negligible for modeling the multi-regime characteristics of the triple flame.

The theory presented in this work can be used for the analysis of fully-resolved numerical datasets of multi-regime combustion phenomena. However, for the method to reach its full potential, a numerical algorithm for solving the two-dimensional composition space equations is required. Therefore, two aspects should be investigated in future works: (1) closures for the gradients of the conditioning variables, which properly incorporate the orthogonality of the coordinates, and (2) identification of a robust numerical solution algorithm for the composition space equations avoiding cross-terms. Besides this, an extension of the composition space equations to incorporate differential diffusion and curvature effects would represent a valuable addition.

Figure 3 :

 3 Figure 3: Schematic of the laminar triple flame. The flame is established by a mixture stratification (imposed mixture fraction gradient) of the fresh gases at the inlet boundary.

Figure 4 :

 4 Figure 4: Scalar fields of temperature, heat release rate (HRR) and mass fractions of the species CO 2 , O 2 , OH and CO for the triple flame.

Figure 5

 5 Figure 5 shows the scalar fields of the conditioning variables Z and Y c . The scalar mixture fraction field illustrates the stratification along the y-direction which remains effective over the whole domain. The reaction progress has a very similar structure as a chemical product species, which is readily anticipated from its definition Y c = Y H 2 O + Y CO 2 . In the top right plot, the source term ωc ⊥ is shown. Furthermore, the figure displays the gradients |∇Z|,

Figure 5 :

 5 Figure 5: Scalar fields of the conditioning variables Z and Y c , the source term ωc ⊥ (top row) together with the gradients |∇Z|, |∇Y c | and |∇ ⊥ Y c | (bottom row). The progress variable is defined as Y c = Y H 2 O + Y CO 2 .

Figure 6 :

 6 Figure 6: Left: angle between ∇Z and ∇Y c (with overlay of white Y c -isolines). Middle: Y c -scalar field with an overlay of gradient trajectories for ∇Z (red) and ∇Y c (white). Gradient trajectories are defined for a gradient field analogously to streamlines for a velocity vector field. Right: Y c -scalar field with an overlay of gradient trajectories for ∇Z (red) and ∇ ⊥ Y c (white).

  ) [J/(kg s)]

10

 10

Figure 7 :

 7 Figure 7: Top: Contour plot of the heat release rate (grey scales) and the distribution of local reaction zones according to the GFRI analysis. Bottom: normalized flame index ξ evaluated from Eq. (40).

Figure 8 :

 8 Figure 8: Normalized budgets of the species equation, Eq. (27), for Y CO along five x-slices (vertical red lines in contour plot). All terms which appear in the 2D composition space equations were post-processed from the detailed chemistry solution of the triple flame.

Figure

  Figure Normalized budgets of the species equation, Eq. (27), for Y CO along several y-slices (horizontal red lines in contour plot). All terms which appear in the 2D composition space equations were post-processed from the detailed chemistry solution of the triple flame.

Figure 10 :

 10 Figure 10: Normalized balances of composition space equations for the species H 2 O, CO 2 , CO and OH. Left column: balance of the flamelet equations for non-premixed (Z-)flamelets, Eq. (34). Middle: balance of the flamelet equations for premixed (Y c -)flamelets, Eq. (30). Right: balance of the 2D composition space equations, Eq. (27). All terms were post-processed from the detailed chemistry solution of the triple flame and then summed up to yield the respective equation balance. Then, the balance values were normalized with the local chemical source term. Points are shown for which the chemical source term of the respective species is larger than 0.1 % of its maximum value in the computational domain.

As stated below Eq. (17), partial derivatives with respect to e are assumed small and omitted.

Acknowledgments

Financial support is kindly acknowledged from the Deutsche Forschungsgemeinschaft(DFG, German Research Foundation) -Projektnummer 325144795.


Kaidi Wan
email: kaidi.wan@coria.fr

Sandra Hartl

Luc Vervisch

Pascale Domingo

Robert S Barlow

Christian Hasse

Combustion regime identification from machine learning trained by Raman/Rayleigh line measurements

Keywords: Convolutional neural network, Combustion regime, Raman/Rayleigh, Flame regime marker, Gradient-free regime identification

Introduction

Complex multi-regime combustion scenarios, including premixed, partially premixed and nonpremixed flames, are commonly reported in combustion systems due to partial premixing of the reactants before they enter the reaction zones, or because of strong finite rate chemistry effects, or even local quenching. The so-called Takeno's flame index [START_REF] Yamashita | A numerical study on flame stability at the transition point of jet diffusion flames[END_REF] is the most popular identifier of local combustion regimes. This flame index exploits information on the degree of alignment between fuel and oxidiser gradients, to determine the nature of the local combustion regime ranging between the two canonical ones, premixed flame (fuel and oxidizer gradients pointing in the same direction) and diffusion flame (gradients in opposite directions). This indicator and its slightly modified versions have been popularised by the analysis of the topological properties of reaction zones in direct numerical simulation (DNS) [START_REF] Mizobuchi | A numerical on the formation of diffusion flame islands in a turbulent hydrogen jet lifted flame[END_REF][START_REF] Domingo | DNS analysis of partially premixed combustion in spray and gaseous turbulent-flame bases stabilized in hot air[END_REF][START_REF] Luo | Direct numerical simulations and analysis of three-dimensional n-heptane spray flames in a model swirl combustor[END_REF] and large-eddy simulation (LES) [START_REF] Domingo | Partially premixed flamelets in LES of nonpremixed turbulent combustion[END_REF][START_REF] Nambully | A Filtered-Laminar-Flame PDF sub-grid scale closure for LES of premixed turbulent flames. Part II: Application to a stratified bluff-body burner[END_REF][START_REF] Ma | Structure of spray in hot-diluted coflow flames under different coflow conditions: A numerical study[END_REF]. Because they require the knowledge of three-dimensional gradients of multiple scalars, these combustion regimes identifiers, and others based on molecular diffusive fluxes estimations [START_REF] Nambully | A Filtered-Laminar-Flame PDF sub-grid scale closure for LES of premixed turbulent flames. Part II: Application to a stratified bluff-body burner[END_REF][START_REF] Domingo | Large-Eddy Simulation of a lifted methane-air jet flame in a vitiated coflow[END_REF][START_REF] Knudsen | A general flamelet transformation useful for distinguishing between premixed and nonpremixed modes of combustion[END_REF], have seldom been extended to experimental measurements [START_REF] Rosenberg | Flame index and its statistical properties measured to understand partially premixed turbulent combustion[END_REF].

Recently, Hartl et al. [START_REF] Hartl | Regime identification from Raman/Rayleigh line measurements in partially premixed flames[END_REF][START_REF] Butz | Local flame structure analysis in turbulent CH4/air flames with multi-regime characteristics[END_REF] proposed a novel combustion regime characterization solely relying on spatial one-dimensional Raman/Rayleigh line measurements of major species concentrations and temperature. In this approach, to complement the knowledge of the Raman/Rayleigh accessible major species and temperature, the full thermochemical state is first approximated by simulating a homogeneous constant-pressure and constant-temperature reactor. After the radicals and minor species have built up, the heat release rate (HRR) is calculated and the chemical explosive mode analysis (CEMA) is performed [START_REF] Lu | Three-dimensional direct numerical simulation of a turbulent lifted hydrogen jet flame in heated coflow: a chemical explosive mode analysis[END_REF]. These two key flame markers are then combined with the mixture fraction (or local equivalence ratio), to identify different combustion regimes, thanks to a gradient-free regime identification (GFRI) strategy [START_REF] Hartl | Regime identification from Raman/Rayleigh line measurements in partially premixed flames[END_REF][START_REF] Butz | Local flame structure analysis in turbulent CH4/air flames with multi-regime characteristics[END_REF]. It was evaluated using DNS of stratified flames [START_REF] Hartl | Assessing an experimental approach for chemical explosive mode and heat release rate using DNS data[END_REF] and applied to a lifted flame experiment [START_REF] Hartl | Assessing the relative importance of flame regimes in Raman/Rayleigh line measurements of turbulent lifted flames[END_REF] and to the Darmstadt multi-regime burner configuration [START_REF] Butz | Local flame structure analysis in turbulent CH4/air flames with multi-regime characteristics[END_REF]. In the latter work, GFRI was extended by an automated procedure to characterize local flame structures based on the relative HRR from premixed and non-premixed reaction zones in close proximity. This gradient-free combustion regime identification has been shown to be robust and accurate. Nonetheless, a quite large amount of CPU time is required to process the measurements and predict the corresponding combustion regime. Indeed, the pointwise approximation of the full thermochemical state, is obtained by numerically converging solutions of homogeneous reactors with a detailed chemical scheme. In addition, the CEMA analysis requires computing the eigenvalues of the chemical Jacobian matrix to get the chemical explosive mode for every point of the one-dimensional line. In the end to fully analyse turbulent jet flame measurements, the overall procedure usually take hours on a regular workstation Machine learning technologies are based on learning from a database to then make predictions. These methods belongs to a rapidly growing field [START_REF] Blasco | Modelling the temporal evolution of a reduced combustion chemical system with artificial neural network[END_REF][START_REF] Franke | Tabulation of combustion chemistry via Artificial Neural Networks (ANNs): Methodology and application to LES-PDF simulation of Sydney flame L[END_REF][START_REF] Aversano | Application of reduced-order models based on PCA & Kriging for the development of digital twins of reacting flow applications[END_REF]. One of the machine learning technologies based on convolutional neural networks (CNNs), which was originally developed for analyzing visual representations [START_REF] Lecun | Deep learning[END_REF][START_REF] Aghdam | Guide to convolutional neural networks[END_REF], has been recently introduced in combustion modeling [START_REF] Nikolaou | Progress Variable Variance and Filtered Rate Modelling Using Convolutional Neural Networks and Flamelet Methods[END_REF][START_REF] Lapeyre | Training convolutional neural networks to estimate turbulent sub-grid scale reaction rates[END_REF][START_REF] Seltz | Direct mapping from LES resolved scales to filtered-flame generated manifolds using convolutional neural networks[END_REF] and flame control [START_REF] Zhu | Convolutional neural network based combustion mode classification for condition monitoring in the supersonic combustor[END_REF]. In the present study, the gradient-free identification regime is combined with CNN, to provide an ultra-fast and reliable combustion regime determination based on thermochemical properties.

The experimental database is detailed in the subsequent section along with the proposed machine learning procedure trained from the GFRI automated classification. Then, the developed approach is tested over two different databases to examine its capability of combustion regime identification.

Methodology

1D Raman/Rayleigh measurement database

One-dimensional Raman/Rayleigh/CO-LIF line imaging has been collected on the multi-regime burner (MRB) configuration with the experimental apparatus detailed in [START_REF] Butz | Local flame structure analysis in turbulent CH4/air flames with multi-regime characteristics[END_REF]. The burner consists of three co-flowing inlet streams, a central jet and two slots (Table 1). The ensemble is protected from the environment by an additional air coflow at 1 m/s. To facilitate laser measurements in the vicinity of the burner, the three inlet streams are inclined by 26 • from each other. The central jet tube, with an inner and outer diameters of 3.0 mm and 3.3 mm, respectively, is surrounded by the first annular slot of 7.0 mm diameter, followed by a second annular slot between 40 mm and 60 mm from the burner centerline. The outer diameter of the burner is 80 mm. Varying the equivalence ratios of the jets, numerous combustion regimes have already been reported [START_REF] Butz | Local flame structure analysis in turbulent CH4/air flames with multi-regime characteristics[END_REF].

Two cases summarized in Table 1 are considered thereafter. Flame MRB18b lies slightly outside the rich flammability limit, while MRB26b includes a large region of mixtures beyond the flammability limit, to promote non-premixed and multi-regime combustion scenarios.

Table 1: Multi-regime burner operating conditions [START_REF] Butz | Local flame structure analysis in turbulent CH4/air flames with multi-regime characteristics[END_REF]. u b (m • s -1 ) is the bulk velocity and φ the equivalence ratio.

Case

Central Single-shot distributions along 6 mm sample lines of species mass fractions, i.e., H 2 , O 2 , CO, CO 2 , CH 4 , H 2 O, and N 2 , and temperature are available with a data spacing of 20 µm for five downstream locations (h = 15, 22.5, 30, 60 and 90 mm) [START_REF] Butz | Local flame structure analysis in turbulent CH4/air flames with multi-regime characteristics[END_REF]. They have been obtained using spontaneous Raman scattering, Rayleigh scattering, and two-photon CO-LIF diagnostics, leading to more than 1000 samples for most measurement locations. These radial profiles were collected by traversing the burner in the direction of the laser axis.

The combustion regimes are first determined at every point of the sampled instantaneous lines via GFRI [START_REF] Hartl | Regime identification from Raman/Rayleigh line measurements in partially premixed flames[END_REF][START_REF] Butz | Local flame structure analysis in turbulent CH4/air flames with multi-regime characteristics[END_REF]. In total six classes are considered: premixed (P), dominantly premixed (DP), multi-regime (MR), dominantly non-premixed (DNP), non-premixed (NP), and lean backsupported (LBS). These are obtained first defining regions of interest from the zero crossing of the chemical mode, then three reference values of the heat release rate within these regions are collected to build a normalized parameter η, used to delineate between the combustion regimes, see [START_REF] Butz | Local flame structure analysis in turbulent CH4/air flames with multi-regime characteristics[END_REF] for all the details on previously published GFRI method.

CNN regime identification approach

While traditional machine learning methods could address this problem point by point, CNN offers the possibility of analyzing the full profile in a single inference. In this so-called "image segmentation" approach, the input line is seen as a whole while the output classifies each pixel, thus preserving the locality of the information.

The CNN input is composed of the one-dimensional profiles, T , of temperature, and Y i , of species mass fractions, H 2 , O 2 , CO, CO 2 , CH 4 , H 2 O and N 2 . The output is the one-dimensional distribution of combustion regime index ξ. The raw one-dimensional profiles have a non-uniform length ranging between 126 and 247 pixels. Various procedures have been tested to secure a uniform CNN input. Better results are obtained by filling the raw profiles with a fixed dummy value, to reach a uniform length of 256 = 2 8 pixels. The dummy value used for the CNN input is zero, i.e., for species mass fractions and temperature, while it is 7 for the corresponding target combustion regime index. The combustion index ξ used here is an integer in the range [0, 7], corresponding to the six regimes, an additional not burning (NB) state and the dummy value, as summarized in Table 2. The samples without combustion over the entire line are not considered.

The resulting numbers of sample lines used for each location of the two flames are summarized in Table 3.

The raw database contains 0.0001% of outlier values, which were dropped out. Then, the CNN operates with normalized quantities, thereby the seven species mass fractions and the tem- In recent years, several architectures have been introduced for the image segmentation tasks, as fully convolutional network (FCN) [START_REF] Long | Fully Convolutional Networks for Semantic Segmentation[END_REF], SegNet [START_REF] Badrinarayanan | SegNet: A Deep Convolutional Encoder-Decoder Architecture for Image Segmentation[END_REF], U-Net [START_REF] Ronneberger | U-Net: Convolutional Networks for Biomedical Image Segmentation[END_REF] and fully convolutional DenseNet [START_REF] Jegou | The One Hundred Layers Tiramisu: Fully Convolutional DenseNets for Semantic Segmentation[END_REF]. In the present study, a U-net inspired architecture is adopted because it provides good performance with a relatively simple network structure.

Originally the U-net is optimized for dealing with two-dimensional images, it is here adapted to one-dimensional information. As shown in Fig. 1, the CNN contains a downsampling (left side) and an upsampling path (right side). One downsampling step consists of two padded onedimensional convolutions with a kernel of 3 (the dimension of the filter is the number of channels of the incoming layer × 3, e.g., at the first downsamping step following the input layer the dimension of the filter for the first convolution is 8 × 3, while that for the second convolution becomes 64 × 3), each followed by a rectified linear unit (ReLU) and a maxpooling operation with stride 2.

The number of feature channels are doubled at each downsampling step. In practice, the number of kernels in each filter is the number of channels of the incoming layer. Then, the number of filters for each convolution is the number of channels of the outcoming layer. Hence, the total number of kernels for each convolution is the product of the number of channels at the incoming layer by the number of channels at the outcoming layer (e.g., at the first downsampling step following the input layer 8 × 64 kernels are used for the first convolution, while 64 × 64 kernels are used for the second convolution).

The upsampling path is a mirrored version of the downsampling path: it includes an upsampling operation which halves the number of feature channels, a concatenation operation which merges the corresponding feature map from the downsampling path, and two padded one-dimensional convolutions, each followed by a ReLU. At the final layer, a convolution with a kernel of 1 is used, with a Softmax activation to classify each 64-component feature vector to one of the eight combusion regimes. Tests have been performed with a single or three convolutions for each downsampling and upsampling step applied to the database, but as usual for the U-Net [START_REF] Ronneberger | U-Net: Convolutional Networks for Biomedical Image Segmentation[END_REF] architecture, the best results are obtained with two convolutions.1 The training of the CNN is performed using the TensorFlow Python library with GPU support (www.tensorflow.org). A series of convolution, maxpooling and upsampling operations are performed iteratively, in which the neural weights are modified until a satisfying minimal departure is obtained between the "target" combustion regime index ξ and the CNNs predictions. The Adam optimizer [START_REF] Kingma | A Method for Stochastic Optimization[END_REF] is applied with a sparse-categorical-crossentropy loss function comparing the prediction against the target.

To relieve overfitting, a drop-out rate of 50% is applied before the last maxpooling and the first upsampling operations (See Fig. 1). Data augmentation is also employed during training, to increase the quality of the results. A mirror operation is first applied to each training sample line, to avoid the emergence of a preferential direction, then each original line now appears twice, in its original form and mirrored. Additionally, an uncorrelated random perturbation of 5% is added to each of these training sample lines, to build a new version of each original and mirrored lines with noise added. The "target" of the noise-added versions remains the same, thus mitigating the risk of overfitting. The final training dataset is the combination of the original, mirrored and noise-added versions, leading to a total of 33924 sample lines feeding the CNN for training.

The training is performed on a batch of 1024 lines, leading to 34 steps per "epoch"/iteration.

The learning rate [START_REF] Aghdam | Guide to convolutional neural networks[END_REF] is initially set to 0.0003 and decays versus the steps (n) according to

l r = 0.0003 1.0 + 0.0001 • n . ( 1
)
The network is trained for 200 epochs to reach convergence, this requires about 45 minutes on an Nvidia GeForce GTX 1080 Ti GPU. The trained CNN reaches a prediction accuracy of 99.3% for the training database. Once trained, returning the combustion regime from an input line is almost instantaneous.

Results and discussion

CNN testing on the remaining location of MRB26b

After being trained on three of the five locations (h = 15, 30 and 90 mm) of the MRB26b dataset (Table 1), the CNN is tested at first on the remaining locations, h = 22.5 and 60 mm, of the same flame, which contains in total 6116 sample lines (Table 3), each with 256 pixels. By comparing, ξ cnn (x), the combustion regime index predicted by the CNN to ξ ref (x), the reference combustion regime index obtained from GFRI [START_REF] Hartl | Regime identification from Raman/Rayleigh line measurements in partially premixed flames[END_REF][START_REF] Butz | Local flame structure analysis in turbulent CH4/air flames with multi-regime characteristics[END_REF], the CNN prediction is found to be 94.3% and 95.0% accurate 2 for h = 22.5 and 60 mm, respectively, which is satisfactory considering the complexity of the problem.

Figure 2 shows representative results for three sample lines from h = 22.5 mm and one from 60 mm of this MRB26b case. For the sample line 1 of h = 22.5 mm (Fig. 2(a)), a first premixed zone 2 The dummy pixels are removed when calculating the accuracy. 1). Left axis: Normalized mass fractions and temperature. Right axis: combustion regime index (Table 2). Circle: GFRI, one of every three pixels is plotted. Line: CNN. by premixed combustion at this location which is not far from the nozzle. Further downstream, up to h = 60 mm (Fig. 3(b)), the multi-regime characteristics become the most significant with a relative population higher than 70%. This is in agreement with the MRB26b flame featuring a large region of mixtures above the rich flammability limit, thus promoting multi-regime combustion events. The lean back-supported regime is not found at this location (h = 60 mm), because it requires support from hot recirculating products, which is more likely to appear in the near-field region close to the nozzle, e.g., h = 22.5 mm (Fig. 3(a)). The predictions of the CNN on the distribution of various combustion regimes agree well with the reference GFRI calculations. The predictions of CNN are compared to the reference results by the GFRI approach [START_REF] Hartl | Regime identification from Raman/Rayleigh line measurements in partially premixed flames[END_REF][START_REF] Butz | Local flame structure analysis in turbulent CH4/air flames with multi-regime characteristics[END_REF]. Red: GFRI. Green: CNN.

CNN testing on the unseen MRB18b dataset

The reliability of the CNN is now evaluated over all locations of the MRB18b dataset, which detected by the machine learning approach, with an accuracy of more than 85% for a dataset unseen during training. This accuracy needs to be put in perspective with the level of detail included in the analysis, which lead to consider six combustion sub-regimes, hence with a refinement that is not possible by most combustion regime identification procedures available so far. Indeed, usually three regimes (premixed, diffusion and partially premixed) only are distinguished by gradient or flux-based regime identification methods. Also, the combustion regime prediction in this study is pixel-wise, while most combustion regime identification methods usually operate over flow zones.

In practice, the CNN can also be used to determine the dominant combustion regime observed in a single shot, or a series of shots defining a flow zone, still benefiting from the high level of confidence provided by GFRI initially used for training.

The training dataset of the present work is much smaller than typical deep learning datasets, even though returning quite accurate predictions. Compared to the most advanced combustion regime identification, as GFRI, the CNN is ultra-fast, making it possible to envision real-time combustion regime identification for advanced flame control, in order to mitigate pollutant emissions occurring under given local combustion regimes.

Once trained for a given range of fuel/air mixtures, it is expected that the neural network can be applied to different burner geometries. However, so far it cannot be expected very precise for free-stream equivalence ratios that would be far from those seen during the training phase.

  perature are normalized by their respective maximum values over the whole database. These maximum values are CO 2 /0.205, O 2 /0.233, CO/0.329, N 2 /0.767, CH 4 /0.074, H 2 O/0.235, H 2 /0.005, T (K)/2293.3.

Figure 1 :

 1 Figure 1: Structure of the CNN for combustion regime identification.

Figure 2 :

 2 Figure 2: Sample 1D lines at two h streamwise positions of MRB26b (Table1). Left axis: Normalized mass fractions and temperature. Right axis: combustion regime index (Table2). Circle: GFRI, one of every three pixels is plotted. Line: CNN.

(ξ = 1 )Figure 3 :

 13 Figure 3: Averaged distributions over sample lines of premixed (P), dominantly premixed (DP), multi-regime (MR), dominantly non-premixed (DNP), non-premixed (NP), and lean back-supported (LBS). (a) At the streamwise position h = 22.5 mm and (b) h = 60 mm of MRB26b. The predictions of CNN are compared to the reference results by the GFRI approach[START_REF] Hartl | Regime identification from Raman/Rayleigh line measurements in partially premixed flames[END_REF][START_REF] Butz | Local flame structure analysis in turbulent CH4/air flames with multi-regime characteristics[END_REF]. Red: GFRI. Green: CNN.

Figure 3

 3 Figure 3 compares the relative populations of combustion regimes over the locations h = 22.5 mm and 60 mm of MRB26b predicted by the CNN and the reference GFRI. The relative population of a combustion regime is computed as the number of pixels in this regime divided by the total number of pixels where combustion occurs, in other words the non burning (NB) and dummy pixels have been removed. At the location h = 22.5 mm (Fig. 3(a)), more than 60% of the reaction zones are identified with premixed characteristics, indicating that the flame is dominated

Figure 4 :

 4 Figure4: Averaged distributions over sample lines of premixed (P), dominantly premixed (DP), multi-regime (MR), dominantly non-premixed (DNP), non-premixed (NP), and lean back-supported (LBS) characteristics over all locations of MRB18b. The predictions of CNN are compared to the reference results by the GFRI approach[START_REF] Hartl | Regime identification from Raman/Rayleigh line measurements in partially premixed flames[END_REF][START_REF] Butz | Local flame structure analysis in turbulent CH4/air flames with multi-regime characteristics[END_REF]. Red: GFRI. Green: CNN.

  contains 18675 sample lines, each with 256 pixels. No location of this flame has been used for training. The predictions of the CNN on the relative populations of P, DP, MR, DNP, NP and LBS characteristics (Table2) are summarized in Fig.4. The flame structures of MRB18b are driven by lean back-supported, multi-regime, and premixed regimes. The CNN predictions achieve an overall good agreement against the GFRI results, with a good representation of the relative distribution of the combustion regimes over the six possibilities.

Figure 5

 5 Figure5shows representative results for three sample lines from four different streamwise positions of the MRB18b case. The lean back-supported zone (ξ = 6) at h = 15 mm (Fig.5(a)), two premixed zones (ξ = 1) at h = 30 mm (Fig.5(b)), and one multi-regime zone (ξ = 3) and one non-premixed zone (ξ = 5) at h = 90 mm (Fig.5(d)) are well captured by the CNN. At h = 60 mm (Fig.5(c)), the CNN predicts a large entire dominatly premixed zone (ξ = 2) instead of two separate ones indicated by the GFRI, but this should be acceptable since the two separte ones are closely adjacent. These profiles have been chosen to illustrate the limitation of the approach observed for a few points. A mismatch is seen at h = 15 mm (Fig.5(a)), where around x =

Table 2 :

 2 Values of the combustion regime index ξ and corresponding regime.

	ξ Combustion regime
	0 Not burning	NB
	1 Premixed	P
	2 Dominantly premixed	DP
	3 Multi-regime	MR
	4 Dominantly non-premixed DNP
	5 Non-premixed	NP
	6 Lean back-supported	LBS
	7 Dummy value

Table 3 :

 3 Number of line samples for the Flames MRB18b and MRB26b at five downstream locations (h) from the nozzle.

	h (mm) MRB18b MRB26b
	15	5646	3514
	22.5	4313	4434
	30	4790	3927
	60	2907	1682
	90	1019	1040

A more complex and advanced structure inspired by the fully convolutional DenseNet[START_REF] Jegou | The One Hundred Layers Tiramisu: Fully Convolutional DenseNets for Semantic Segmentation[END_REF] has also been tested, without much improvement in prediction accuracy, but with a significantly higher computational cost.

Acknowledgments

Financial support is kindly acknowledged from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) -Projektnummer 325144795. The first author is supported by ADEME (Agence de l'Environnement et de la Maîtrise de l'Energie, France) under the project IGAR (Injection de Gaz de Recycle) Grant Nb 1882C001. Computing resources were provided by CRIANN (http://www.criann.fr).

 1). Left axis: Normalized mass fractions and temperature. Right axis: combustion regime index (Table 2). Circle: GFRI, one of every three pixels is plotted. Line: CNN.

85 pixels of the line sample, the GFRI indicates a multi-regime zone, which is not detected by the CNN. Besides, at the x = 139 pixels of the line sample at h = 30 mm (Fig. 5 By comparing the combustion regime index predicted by the CNN, ξ cnn (x), to the reference combustion regime index, ξ ref (x), determined by the GFRI approach [START_REF] Hartl | Regime identification from Raman/Rayleigh line measurements in partially premixed flames[END_REF][START_REF] Butz | Local flame structure analysis in turbulent CH4/air flames with multi-regime characteristics[END_REF] at every pixel of the 18675 sample lines, 86.1% of prediction accuracy is found for this quite challenging complex multi-regime MRB18b dataset unseen during training. The CPU time required for CNN prediction is of 2 s for the entire MRB18b dataset. Figure 6 displays the accuracy of the CNN over each of the five downstream locations of the MRB18b case. A precision of more than 85% is achieved for every location, which confirms the CNN accuracy.

Conclusions

A convolutional neural network (CNN) has been trained to determine combustion regimes in a methane-air turbulent flame from Raman/Rayleigh one-dimensional measurements of major species (H 2 , O 2 , CO, CO 2 , CH 4 , H 2 O, and N 2 ) and temperature. The experimental data from a recently investigated multi-regime laboratory scale burner [START_REF] Butz | Local flame structure analysis in turbulent CH4/air flames with multi-regime characteristics[END_REF] are first processed with the gradient free regime identification (GFRI) method [START_REF] Hartl | Regime identification from Raman/Rayleigh line measurements in partially premixed flames[END_REF][START_REF] Butz | Local flame structure analysis in turbulent CH4/air flames with multi-regime characteristics[END_REF], to generate a pixel-wise combustion regime database.

The CNN is trained at various heights along the burner axis for one operating condition and tested at two locations not included in the training. Then, the CNN is applied to a different burner operating condition, for which the free-stream equivalence ratios have been modified. Six classes of combustion regimes, including premixed (P), dominantly premixed (DP), multi-regime (MR), dominantly non-premixed (DNP), non-premixed (NP), and lean back-supported (LBS), are well