1 Introduction

In many technically relevant combustion devices, flames develop in conditions where fuel and oxidizer are not perfectly mixed. These partially premixed or mixed-mode flames are challenging for numerical simulations due to the co-existence of premixed and non-premixed combustion characteristics. They have, however, the potential to increase stability and reduce pollutant formation. Over the last decades, different approaches have been developed to define markers which can identify different combustion regimes, either from simulation data or measurements.

Historically, the first flame regime marker was developed by Yamashita et al. (1996) and is based on the alignment of gradients of the fuel and oxidizer mass fractions. In the last few years alone, a large number of works has used the concept of the Takeno flame index to study flames and improve their prediction: it was used to understand the structure of blue swirling flames (Chung et al. 2019), identify flame regimes in thickened flame simulations of spray flames (Hu and Kurose 2019b; Dressler et al. 2020) and study the influence of evaporation (Wei et al. 2018) as well as devolatilization (Zhang et al. 2017). Premixed and non-premixed regions in swirl spray flames were analyzed with the flame index concept (Eckel et al. 2019; Paulhiac et al. 2020) and the influence of swirl number on the combustion regime (Fredrich et al. 2019) was evaluated. It was used to study different coal flames (Tufano et al. 2018; Wan et al. 2019; Rieth et al. 2019; Akaotsu et al. 2020), assess the stabilization mechanism of supersonic lifted flames (Bouheraoua et al. 2017) and lifted hydrogen flames (Benim et al. 2019), classify regions of industrial burners by the combustion regime (Zhang et al. 2020), study auto-ignition of supercritical hydrothermal flames (Song et al. 2019), analyze extinction and re-ignition events (Sripakagorn et al. 2004) and gain insight into partially premixed DME flames (Hartl et al. 2019). It was further applied to scram jets (Shen et al. 2020; Zhao et al. 2019), Moderate or Intense Low oxygen Dilution (MILD) combustion (Amaduzzi et al. 2020), compression ignition engines (An et al. 2018), double-cone burners (Zhao et al. 2019), coaxial diffusion flames (Akaotsu et al. 2020), edge flames (Duboc et al. 2019) and jet-in-hot-coflow flames in conjunction with tangential stretching rate analysis (Li et al. 2019). In order to evaluate the flame index experimentally, Rosenberg et al. (2013, 2015) introduced tracer gases. This large number of works in just the last couple of years shows that the flame index and its variations (Fiorina et al. 2005; Lock et al. 2005) based on the alignment of species gradients are actively used to study many different types of flames (Masri 2015). It is therefore important to understand the behavior of this marker when applied to partially premixed flames.

Although the flame index is the most commonly used flame regime marker due to its simplicity, other markers have been developed as well. Based on flamelet analysis, the gradient alignment of mixture fraction and reaction progress variable has been discussed to determine the combustion regime (Favier and Vervisch 2001; Domingo et al. 2002; Nguyen et al. 2010; Lamouroux et al. 2014; Hu and Kurose 2019a). A similar approach can be used to define Damköhler numbers for the premixed, non-premixed and partially premixed regime Domingo et al. (2008) or compare source terms for premixed and non-premixed flamelets (Knudsen and Pitsch 2009, 2012). Another approach uses the chemical explosive mode analysis (CEMA) to differentiate between premixed propagation and diffusion flame characteristics. For example, Lu et al. (2008) compared the flame index to the chemical modes in a turbulent hydrogen flame and Doan and Swaminathan (2019) compared the regime prediction of the flame index with the chemical mode in MILD combustion and concluded that the CEMA based approach identified large parts of the flame as premixed whereas the flame index identified a non-premixed regime. Nordin-Bates et al. (2017) applied a similar concept to a scram jet flame. Hartl et al. (2018, 2019, 2019) and Butz et al. (2019) developed a flame regime marker known as Gradient Free Regime Identification (GFRI) which uses the chemical mode and allows to compute the marker from experiential Raman/Rayleigh spectroscopy results. It has been successfully used to train convolutional neural networks (Wan et al. 2020) for regime identification. Another approach was taken by Wu and Ihme (2015). They developed a marker (Wu and Ihme 2016) that evaluates a drift term. The drift term measures how good the prediction of a set of manifold-based combustion models fits to the current flow field conditions for a set of quantities of interest (QoI). A detailed description of the definition of these flame regime markers and their development is given in the “Appendix”.

All aforementioned regime markers have been derived from different assumptions and have their limitations. For example, the flame index based on fuel and oxidizer gradients is not applicable in regions of the flame where the fuel is completely consumed. Because of this, different multi-species extensions have been used in the past (Lignell et al. 2011; Jangi et al. 2015; Minamoto et al. 2015; Ma et al. 2016; Pei et al. 2016; Wei et al. 2018; Pang et al. 2018; Tufano et al. 2018; Hu and Kurose 2019b; Wan et al. 2019; Wen et al. 2019; Akaotsu et al. 2020). Fiorina et al. (2005) found that even in laminar diffusion flames, the flame index concept can make wrong predictions. They therefore introduced the oxygen gradient of a premixed model flame as an additional parameter. The regime markers based on flamelet analysis have their limitations when used in direct numerical simulation of turbulent flames because instantaneous flame fronts are subjected to unsteady effects, curvature, flame stretch and the direct interaction of different combustion regimes. Different approaches are available to overcome this by incorporating two-dimensional composition space equations (Scholtissek et al. 2020), accounting for curvature effects (Scholtissek et al. 2017) and analyzing the effect of tangential and preferential diffusion (Dietzsch et al. 2018; Han et al. 2019). The drift term approach by Wu and Ihme is not necessarily a marker for premixed and non-premixed regions. Instead, it shows how close the trajectory of a quantity of interest, e.g. a species mass fraction, on the manifold of a combustion model is to the local flow field conditions. It therefore depends on the choice of QoI and the selected combustion models. While partially premixed flames are generally characterized by a mix of premixed and non-premixed characteristics in a compositionally inhomogeneous flow field (Peters 2001), there is no universally accepted formal definition of the partially premixed combustion regime, especially in turbulent flames due to the complexity of the underlying physics (Wu and Ihme 2016; Hartl et al. 2019). Effects like back-supported flame fronts, time history effects of equivalence ratio fluctuations and diffusion of intermediate species in adjacent flame fronts make partially premixed flames fundamentally different from the simple premixed and non-premixed model flames (Lipatnikov 2012). A simple differentiation into premixed and non-premixed can therefore lead to misleading results (Wu and Ihme 2016). Because of this, it cannot be expected that any flame regime marker is able to identify combustion regimes unambiguously in every situation (Wu and Ihme 2016). Nonetheless, using regime markers is important for advancing the understanding of flames and phenomena like flame stabilization as described above. Some combustion models also use regime markers to blend between premixed and non-premixed combustion submodels in the case of partially premixed combustion situations (Domingo et al. 2002; Knudsen and Pitsch 2012; Yadav et al. 2017; Hu and Kurose 2019b, a; Hu et al. 2020).

Because of this, the goal of this work is to provide a side-by-side comparison of different flame regime markers in terms of their ability to classify the combustion regime in a simulation of a turbulent partially premixed flame, where the flame is fully resolved. To the best knowledge of the authors, a direct comparison of all different regime marker concepts in a compositionally inhomogeneous partially premixed flame has not been done before. The flame setup is the Sydney/Sandia flame with inhomogeneous inlets (Meares and Masri 2014; Meares et al. 2015; Barlow et al. 2015) in configuration FJ-5GP-Lr75-57. It is a well documented and experimentally investigated partially premixed turbulent flame of laboratory scale. A retractable central fuel pipe allows adjusting the degree of premixing of fuel and air, thus creating inhomogeneous conditions. An early simulation of this configuration was performed by Wu and Ihme (2016). They performed two LES, one with a reduced-manifold combustion model based on premixed flames (premixed filtered tabulated chemistry LES, FTACLES) and one based on non-premixed flames (flamelet/progress-variable, FPV). Since these models are derived assuming perfectly premixed or non-premixed flames, they cannot describe flames where no single combustion regime dominates. By introducing a drift term, which measures how well a given manifold-based combustion model can represent the flame based on the current flow field, they defined a new regime marker. This marker compares the drift term from the premixed manifold combustion model to that of the non-premixed model. The authors observed that regime markers based solely on major species are insufficient to describe the complex processes in multi-regime combustion and therefore are not suited to blend the two combustion models for improving profiles of intermediate species. In contrast to other regime markers, their drift-term based marker is therefore species-specific and marks regions where a particular species mass fraction is highly sensitive to the manifold representation. They also found that major species can be largely insensitive to the combustion model, while minor species like CO are highly sensitive. Kleinheinz et al. (2017) performed LES with a multi-regime combustion model, which computes the contributions from premixed and non-premixed flamelets and uses this information to blend two combustion models, in order to study the stabilization mechanism of the flame. They found that the high stability of the Lr75 case is due to the high heat release rate in the premixed domain directly at the nozzle. Maio et al. (2017) conducted LES with tabulated chemistry and thickened flame models, and were able to capture radial species profiles from the experiment near the nozzle, but observed large deviations further downstream. Perry et al. (2017) conducted LES with a flamelet model that utilizes two mixture fractions. This approach significantly increased the accuracy of the results compared to using only one mixture fraction for the cases with inhomogeneous inlets and showed that the combustion near the nozzle is predominantly premixed. They also assessed the influence of subgrid filter probability density functions (Perry and Mueller 2019) for this flame. Ji et al. (2018) used a RANS-based multi-environment probability density function (MEPDF) approach, which was able to yield good agreement with the measurements but had some shortcomings in predicting local extinction events downstream. By using a joint composition-enthalpy transported PDF method, Tian and Lindstedt (2019) confirmed that the mixture fraction and reaction progress variable are strongly correlated and that the flame in the inhomogeneous case is predominantly premixed and becomes more diffusion controlled due to the entrainment of the pilot downstream. Chen et al. (2020) showed that an LES model based on unstrained premixed flamelets is able to qualitatively predict the correct trend of local extinction with inflow velocity for the Sydney flame further downstream. Luo et al. (2020) applied a dynamic second-order moment closure model to LES and found that the premixed flame is dominant upstream, causing 70 % of the heat release, which then decreases to about 40 % downstream, with an overall good agreement with measurements. Hansinger et al. (2020) performed LES with the Eulerian Stochastic Fields Method and assessed the influence of the number of stochastic fields on the quality of the simulation results. Shrotriya et al. (2020) conducted LES with reaction diffusion manifolds (REDIM) because it is mostly independent of the combustion regime. They used the flame index to confirm the aforementioned structure of the flame. Due to the high Reynolds number of this setup, it is not possible to simulate the whole flame with a direct numerical simulation. Instead, in this work, data from a quasi-DNS (Zirwes et al. 2020) is used which fully resolves the instantaneous flame fronts and the flow down to the Kolmogorov length in the upstream region (\(x/D<20\)).

The paper is structured as follows: Sect. 2 gives a brief overview of the simulation setup and summarizes the different regime markers used in this work. Section 3 presents the flame structure in terms of averaged heat release rates conditioned to the flame index and statistics of the scalar dissipation rate, mixture fraction and progress variable. A direct comparison of the different regime markers is performed in Sect. 4 on instantaneous cutting planes and radial lines through the flame. In the “Appendix”, different regime markers from the literature are discussed in detail.

2 Numerical Setup

2.1 Simulation Setup

In a previous work (Zirwes et al. 2020), a quasi-Direct Numerical Simulation (quasi-DNS or qDNS) of the Sydney/Sandia flame in configuration FJ-5GP-Lr75-57 has been conducted. The term quasi-DNS has been proposed in the literature (Enger et al. 2000; Nikitin et al. 2000; Touil et al. 2002; Kimura et al. 2002; Sukegawa et al. 2003; Jung and Yoo 2004; Tafti 2005; Raufeisen et al. 2008; Shams et al. 2012; Komen et al. 2014; Mayrhofer et al. 2015; Addad et al. 2015; Chu et al. 2016; Lecrivain et al. 2016; Forooghi et al. 2017; Komen et al. 2017; Zhao et al. 2018; Saeedipour et al. 2019; Wan et al. 2019) to describe a model-free simulation which does not fulfill all of the strict requirements of a DNS, e.g. for complex geometries which do not allow the use of higher order discretization schemes, but is accurate enough so that the results of a full DNS are expected to be similar. In the context of this work, the quasi-DNS resolves the flow field down to the Kolmogorov length and fully resolves the instantaneous flame fronts with at least 20 points while utilizing finite rate chemistry and detailed molecular diffusion. Therefore, no turbulence or combustion models are required. However, because of the large dimensions of this flame, this resolution is only achieved upstream near the nozzle for \(x/D<20\), D = 7.5 mm. Here, x is the direction of the mean flow. Fourth order interpolation schemes for flux reconstruction of the spatial discretization schemes and second order time discretization schemes are employed. Because of this, the term quasi-DNS is chosen to describe this simulation. The accuracy of the employed discretization schemes has been justified by comparing canonical test cases with spectral DNS codes in Zirwes et al. (2020). Running the simulation on a coarser mesh would mean that the flame front cannot be fully resolved anymore, so that the regime markers would have to be evaluated on filtered quantities. Also, additional sub-grid models would be required to recover the correct flame structure, which would interfere with the evaluation of the different types of regime markers in a model-free way.

The full description of the simulation setup is given in Zirwes et al. (2020). Here, the simulation setup is described briefly. In configuration FJ-5GP-Lr75-57 (Barlow et al. 2015) of the Sydney/Sandia flame, the central fuel pipe is retracted by 7.5 cm, thus creating compositionally inhomogeneous conditions. Methane/air mixture enters the domain with a bulk velocity of 57 m/s. The simulation consists of three consecutive parts: At first, the fully developed pipe flow of the central methane pipe and annular air flow is simulated using highly resolved LES. The second simulation is a quasi-DNS of the non-reactive mixing of methane and air; and lastly the reactive quasi-DNS of the flame is performed. The simulations have been conducted with an in-house solver (Zhang et al. 2015, 2016; Zirwes et al. 2018, 2019, 2020) that has been validated with and applied to different flame setups (Zhang et al. 2017, 2017; Zirwes et al. 2019; Zhang et al. 2019; Zirwes et al. 2020; Zhang et al. 2020; Zirwes et al. 2020). It uses OpenFOAM (Weller et al. 2017) to solve the conservation equations for mass, momentum, energy and chemical species in compressible formulation with the finite volume method. Cantera (Goodwin et al. 2017) is used for computing detailed transport properties with the mixture-averaged transport model (Kee et al. 2005). A reaction mechanism by Lu and Law (2008) is used for the oxidation of methane, which contains 19 species, 11 quasi steady-state species and about 200 chemical reactions. The computational meshes for both qDNS (non-reactive mixing and the flame) consist of 150 million hexahedral cells each, with a smallest resolution of 5 μm for the non-reactive mixing and 10 μm for the reactive simulation, which is sufficient to resolve the flame front and the flow down to the Kolmogorov scale in the near-nozzle region. Comparison of the qDNS results with experimental data in Zirwes et al. (2020) shows a quantitatively good agreement in terms of time mean and root-mean square values of temperature and species concentrations as well as instantaneous scatter plots. For more detailed information on the grid resolution validation, detailed comparison to experimental data and general numerical setup, see Zirwes et al. (2020).

2.2 Regime Markers

Table 1 The different types of flame regime markers used in this work

Objective of this work is to utilize the quasi-DNS database of the compositionally inhomogeneous partially premixed Sydney/Sandia flame to evaluate different types of flame regime markers from literature as a post-processing step and to provide a side-by-side comparison of their behavior. In the literature, a direct comparison of all these markers applied to a partially premixed turbulent flame is not available. Because of the qDNS nature of the results and because the reaction zones are fully resolved, the regime markers can be evaluated without modeling assumptions.

In total, three different kinds of flame regime markers are evaluated in this work from the quasi-DNS database and summarized in Table 1.

  • The first type of regime markers uses the alignment of gradients. We evaluate the classical Takeno flame index \(\xi _{{\mathrm {Takeno}}}\) from the mass fractions Y of CH\(_4\) and O\(_2\) as well as its modifications: \(\xi ({{\mathrm {CH}}}_4,{\mathrm {O}}_2)\), which is the normalized Takeno flame index, \(\xi ({\mathrm {CO}},\mathrm{O}_{\mathrm {2}})\) the normalized flame index using CO instead of CH\(_4\) and \(\xi ({{\mathrm {Multi}}},{\mathrm {O}}_2)\) where the fuel species is replaced by the sum \(Y_F=Y_{{{\mathrm {CH}}}_4}+Y_{{\mathrm {CO}}}+Y_{{\mathrm {C}}_2{\mathrm {H}}_2}+Y_{{\mathrm {H}}_2}\). The flame index by Fiorina et al. \(\xi _{{\mathrm {Fiorina}}}\) additionally considers the magnitude of the oxygen gradient and compares it to that in an unstretched premixed flame \(D_O\).

  • The second type of flame regime marker \(\xi _{{\mathrm {GFRI}}}\) uses the gradient free regime identification (GFRI) method based on chemical explosive mode analysis (CEMA) and compares the largest heat release rate in the premixed part of the flame front \(\dot{Q}^{{\mathrm {PF}}}\) to the largest heat release rate peak in the non-premixed part of the flame \(\dot{Q}^{{\mathrm {DF}}}\) along a line.

  • The third marker \(\xi _{{\mathrm {Drift}}}\) computes drift terms which describe how well a given manifold-based combustion model can represent the local flow and flame conditions. The drift term in this work is evaluated in terms of \(D^m={\mathrm {D}}Y_{{\mathrm {CO}}}/{\mathrm {D}}t - {\mathrm {D}}Y_{{\mathrm {CO}}}^m/{\mathrm {D}}t\), which is the value of the substantial derivative of the CO mass fraction in the simulation minus the substantial derivative on the manifold given by combustion model m. The smaller this term, the better can the given model m represent the local conditions. We compare two manifolds: one constructed from steady premixed flames (\(m={\mathrm {PF}}\)) and one constructed from steady diffusion flames (\(m={\mathrm {DF}}\)). As the authors of the drift term analysis note, mass fractions of major species might not be sensitive to the type of manifold, but minor species are. Therefore, we use CO in this study to evaluate the drift terms.

All markers are summarized in Table 1 and defined so that 0 represents the non-premixed regime and 1 the premixed regime (except for the original \(\xi _{{\mathrm {Takeno}}}\) definition). More details about the markers are given in “Appendix A”.

As shown in the introduction, combustion regime markers are widely used to analyze the structure of flames, study phenomena like flame stabilization or flashback, and can aid combustion model selection to improve the accuracy of simulations. Although they have become an important tool for examining flames, different types of flame regime markers can disagree when assigning premixed or non-premixed characteristics to different parts of a flame because they have been derived with varying assumptions and goals. Applying the regime markers to partially premixed turbulent flames for identifying premixed, non-premixed and partially premixed characteristics constitutes an additional problem: Turbulent flames are subject to three-dimensional flame stretch and transient effects like (re-)ignition, which further distorts a clear identification of combustion characteristics in terms of model flames. In addition, since there is no universal formal definition of the partially premixed combustion regime, identifying this regime is mostly done qualitatively. Because of this and due to the complex nature of the physical processes governing partially premixed flames, it cannot be expected that a single regime marker can identify combustion regimes consistently in every situation. It is therefore all the more important to understand the behavior of different regime markers when applied to turbulent partially premixed flames.

Some of the markers from Table 1 require the evaluation of mixture fraction Z and reaction progress c. In this work, the mixture fraction Z is evaluated in terms of the Bilger mixture fraction (Bilger 1979) from the instantaneous species mass fractions:

$$\begin{aligned} Z=\frac{\frac{2(Y_{{\mathrm {C}}}-Y_{{\mathrm {C,Ox}}})}{M_{{\mathrm {C}}}}+\frac{Y_{{\mathrm {H}}}-Y_{{\mathrm {H,Ox}}}}{2M_{{\mathrm {H}}}}-\frac{Y_{{\mathrm {O}}}-Y_{{\mathrm {O,Ox}}}}{M_{{\mathrm {O}}}}}{\frac{2(Y_{{\mathrm {C,F}}}-Y_{{\mathrm {C,Ox}}})}{M_{{\mathrm {C}}}}+\frac{Y_{{\mathrm {H,F}}}-Y_{{\mathrm {H,Ox}}}}{2M_{{\mathrm {H}}}}-\frac{Y_{{\mathrm {O,F}}}-Y_{{\mathrm {O,Ox}}}}{M_{{\mathrm {O}}}}} \end{aligned}$$

Here \({\mathrm {C}}\), \({\mathrm {H}}\) and \({\mathrm {O}}\) denote the elements, \(M_i\) is the atomic weight of element i and the subscripts \({\mathrm {F}}\) and \({{\mathrm {Ox}}}\) denote the fuel and oxidizer stream composition.

For the reaction progress variable \(Y_c\), two common definitions are used in this work (Pierce and Moin 2004) and computed in a post-processing step from the simulation data:

$$\begin{aligned} Y_{c_1} & \equiv {} Y_{{\mathrm {CO}}_2}+Y_{{\mathrm {CO}}} \end{aligned}$$
$$\begin{aligned} Y_{c_2} & \equiv {} Y_{{\mathrm {CO}}_2}+Y_{{\mathrm {CO}}}+Y_{{\mathrm {H}}_2{\mathrm {O}}}+Y_{{\mathrm {H}}_2} \end{aligned}$$

Because \(Y_{c_1}\) and \(Y_{c_2}\) are the sum of species mass fractions, they are not normalized. Therefore, a normalized reaction progress variable (Bray et al. 2005) is introduced as

$$\begin{aligned} c\equiv \frac{Y_c}{Y_c^{{\mathrm {Eq}}}(Z)} \end{aligned}$$

where \(Y_c\) is either \(Y_{c_1}\) or \(Y_{c_2}\). \(Y_c^{{\mathrm {Eq}}}(Z)\) is the value of \(Y_c\) of a mixture with mixture fraction Z at chemical equilibrium, computed from the fresh gas composition at Z, \(T=300\) K, \(p=1\) atm and assuming constant pressure and enthalpy. In this work, \(Y_c^{{\mathrm {Eq}}}(Z)\) has been tabulated for a total number of \(5\cdot 10^4\) mixture fraction values linearly spaced between \(0<Z<1\) for both \(c_1\) and \(c_2\) and interpolated for all cells in the computational domain.

3 Overview of the Flame Structure

Before applying the different regime markers from Table 1, the general flame structure of the Sydney/Sandia flame from the qDNS is discussed in this section. In order to classify the flame, the simulation domain is subdivided into slices with an axial width of \(\varDelta X=10\) mm. The instantaneous volume integrated heat release rate \(\dot{Q}_V\) in each slice is computed by

$$\begin{aligned} \dot{Q}_{V}^{{\mathrm {PF}}}(X)= & {} \int _{X}^{X+\varDelta X} \int _{-\infty }^{+\infty } \int _{-\infty }^{+\infty } \langle \dot{Q},\xi ({{\mathrm {CH}}}_4,{\mathrm {O}}_2)>0.5\rangle \, {\mathrm {d}}z\,{\mathrm {d}}y\,{\mathrm {d}}x \end{aligned}$$
$$\begin{aligned} \dot{Q}_{V}^{{\mathrm {DF}}}(X)= & {} \int _{X}^{X+\varDelta X} \int _{-\infty }^{+\infty } \int _{-\infty }^{+\infty } \langle \dot{Q},\xi ({{\mathrm {CH}}}_4,{\mathrm {O}}_2)<0.5\rangle \, {\mathrm {d}}z\,{\mathrm {d}}y\,{\mathrm {d}}x \end{aligned}$$

which is then averaged over ten uncorrelated time steps. The notation \(\langle \dot{Q},\xi ({{\mathrm {CH}}}_4,{\mathrm {O}}_2)>0.5\rangle\) means that only values of the total heat release rate \(\dot{Q}\) are considered, where the flame shows premixed characteristics. In this integration, all values of \(\dot{Q}\) are considered without conditioning to high heat release rates. \(\dot{Q}_{V}^{{\mathrm {PF}}}\) represents the contribution to total heat release rate from regions with premixed combustion mode. The total heat release rate itself is computed from

$$\begin{aligned} \dot{Q}=-\sum _i\dot{\omega _i}\varDelta h^\circ _i \end{aligned}$$

where \(\dot{\omega }_i\) is the reaction rate of the i-th species and \(\varDelta h^\circ _i\) its enthalpy of formation. The results are depicted in Fig. 1. Black lines show how much heat in total is released depending on the axial position x/D, where D is the diameter of the nozzle, and flame regime (solid line for premixed and dashed for non-premixed). Up to \(10\,D\) near the nozzle, heat is mainly released by premixed regions. Up to \(x/D=2\), heat is released predominantly from lean regions and in the range of \(2<x/D<10\) predominantly from rich regions. Further downstream, total heat release per cross section increases because the reaction zones become generally broader downstream as the flame transitions from a predominantly premixed toward the partially premixed combustion mode. Contributions from both premixed and non-premixed combustion become approximately equal, indicating that both combustion modes are equally important in this mixed-mode flame. Similar results have been found by (Kleinheinz et al. 2017; Perry et al. 2017; Tian and Lindstedt 2019). Luo et al. (2020) found in their LES, that about 70 % of the heat release rate originates from premixed-dominated regions in the upstream part of the flame and further downstream this reduces to 40 %. The quasi-DNS results show that the relative importance of the premixed regions reaches up to 90 % at \(x/D\approx 5\) and stays dominant in the whole range of \(0<x/D<10\). After that, the relative importance of both regions as measured by the heat release rate is about 50 %.

Fig. 1
figure 1

Total heat release rate (black) integrated over axial slices for premixed (solid line) and non-premixed (dashed line) regimes. Also shown is the heat released by the reaction of three species (colored lines)

Fig. 2
figure 2

Scatter plot of temperature zoomed to the high temperature range over mixture fraction at \(x/D=15\) colored by the low range of scalar dissipation rate

Additionally, the qDNS dataset allows to inspect the heat released from the reaction of each species \(\dot{Q}_i=-\dot{\omega }_i\varDelta h^\circ _i\). Here, the highest contribution to the heat release rate comes from \(\hbox {H}_2\hbox {O}\), \(\hbox {CO}_2\) and H. In the upstream region \(x/D<10\), the product species \(\hbox {H}_2\hbox {O}\) and \(\hbox {CO}_2\) are only produced in premixed regions. H radicals are mainly formed endothermically in non-premixed regions and react exothermically in premixed regions.

The scalar dissipation rate \(\chi _{Z}\) (see Eq. (23) in the “Appendix”) is evaluated using Bilger’s mixture fraction. Its effect on the flame can be seen from Fig. 2. The scatter plot of instantaneous temperature over Bilger mixture fraction at the axial position \(x/D=15\) is colored by the scalar dissipation rate in the range of \(0<\chi _Z<1\,\hbox {s}^{-1}\). The diffusion coefficient for the scalar dissipation rate is assumed to be approximately equal the molecular thermal diffusivity of the mixture. As the scalar dissipation rate increases, the peak temperature decreases, which is consistent with the flamelet assumptions.

Fig. 3
figure 3

Joint probability density function (JPDF) at the position \(x/D=15\), \(r/D=0.75\) of reaction progress variable \(Y_c\) from Eq. (2) and (3) as well as their normalized versions from Eq. (4) and Bilger mixture fraction Z. All JPDFs are conditioned to \(\dot{Q}>10^7\) \(\hbox {W/m}^3\). The center and right column are additionally conditioned to \(\xi ({{\mathrm {CH}}}_4,\,{\mathrm {O}}_2)=1\) (premixed) and 0 (non-premixed)

The structure of the flame at \(x/D>10\) can be understood better by looking at the joint probability density function (JPDF) of mixture fraction and reaction progress. Figure 3 shows the JPDF of the reaction progress \(Y_c\) and normalized reaction progress c at the position \(x/D=15\), \(r/D=0.75\). Additionally, the data is conditioned to only show data points where \(\dot{Q}>10^7\) \(\hbox {W/m}^3\). The first column shows a strong correlation between mixture fraction and reaction progress. This correlation is qualitatively the same for \(c_1\) and \(c_2\). The strong correlation has also been noted by Barlow et al. (2017) and Tian and Lindstedt (2019). Therefore, the commonly made assumption of statistical independence of Z and c does not hold in this partially premixed flame. By conditioning the data to the flame index based on the gradient of fuel and oxidizer species \(\xi ({{\mathrm {CH}}}_4,{\mathrm {O}}_2)\) (see Table 1) in the second and third column, the flame structure can be seen to generate its heat from rich premixed (\(\xi ({{\mathrm {CH}}}_4,{\mathrm {O}}_2)=1\), \(Z>0.055\)) and stoichiometric non-premixed regions.

Fig. 4
figure 4

Histogram of the distribution of \(\xi (Z,c_1)\) at \(x/D=15\) from all points of an instantaneous time step with \(\dot{Q}>10^7\) \(\hbox {W/m}^3\). 1 means parallel alignment of mixture fraction and reaction progress variable gradients, 0 is anti-parallel

It is interesting to note that the alignment of mixture fraction and reaction progress variable gradients are mostly parallel or anti-parallel. Figure 4 shows histograms of the cosine of the angle between gradients of Z and c sampled at \(x/D=15\) for regions with \(\dot{Q}>10^7\) \(\hbox {W/m}^3\). This bimodal distribution is true for all different axial positions \(0<x/D<30\). As suggested by Domingo et al. (2002), Favier and Vervisch (2001) and Nguyen et al. (2010), a value of \(\xi (Z,c)\) between 0 and 1 suggests a partially premixed flame. Although the flame shows clear characteristics of partially premixed combustion (see the next section for more details) for \(x/D\ge 10\), the distribution of \(\xi (Z,c)\) is bimodal at 0 and 1. Wu and Ihme (2016) found similar results and concluded that the flame regime marker based on the alignment of Z and c is not able to correctly predict the partially premixed characteristic of this flame. It should be noted, however, that the mixture fraction Z in this work is computed as a post-processing step from the simulation results. Because the simulation considers preferential diffusion, Z is not a conserved scalar here. Nonetheless, the results are consistent to the findings in Wu and Ihme (2016).

4 Direct Comparison of Flame Regime Markers

4.1 Visualization of the Flame Structure

Fig. 5
figure 5

Instantaneous axial cutting planes of: a temperature; b fuel mass fraction gradient; c chemical mode \({{ CM }}\); d non-normalized Takeno flame index; e normalized Takeno flame index; f flame index by Fiorina using \(c_1\) as progress variable; g flame index based on CO and oxidizer. Cutting planes eg only show regions where \(\dot{Q}>10^7\,\hbox {W/m}_{3}\)

In order to get a better overview of the flame, Fig. 5 presents axial cutting planes \(0<x/D<30\) of different quantities for one instantaneous time step, see Table 1. At the top (Fig. 5a), the temperature field is depicted. On the left, the cold, unburnt and partially premixed methane/air enters the domain from the central nozzle. It is surrounded by a hot pilot flow with the composition of methane/air at an equivalence ratio of unity at chemical equilibrium.

In Fig. 5b, the magnitude of the gradient of fuel mass fraction \(Y_{{{\mathrm {CH}}}_4}\) is shown for the same time step as Fig. 5a. The flow enters the domain in a partially premixed state and creates stratified conditions throughout the whole domain. Figure 5c shows the chemical mode \({{ CM }}\) (see Eq. (30)). Figure 5d depicts the non-normalized Takeno flame index from Eq. (9). Positive values mark premixed regions of the flame and negative ones non-premixed regions. In this picture, the Takeno flame index is shown at every location, e.g. also within the central nozzle, where only inert mixing occurs. In contrast to this, the Fig. 5e–g are conditioned to the heat release rate. The gray areas indicate regions where \(\dot{Q}<10^7\) \(\hbox {W/m}^3\), so that the flame index is only shown where an actual flame front is present. Figure 5e shows the normalized Takeno flame index based on the mass fraction of the fuel species and \(\hbox {O}_2\). Upstream near the nozzle in Fig. 5e, the reaction zone is thin and in the premixed regime, as indicated by \(\xi ({{\mathrm {CH}}}_4,{\mathrm {O}}_2)\) being close to unity (red color). On the outside, two additional reaction zones appear which are a consequence of the hot pilot mixing with the surrounding cold air co-flow, resulting in dissociation reactions. Further downstream, the inner part of the flame front is predicted to be premixed with an outer, non-premixed layer. In the last third of the domain, a triple flame structure is visible, with two premixed reaction zones surrounding a non-premixed one. This structure is discussed in more detail further below.

The regime marker by Fiorina et al. from Eq. (16) in Fig. 5f shows similar results to 5e. Here, \(c_1\) from Eq. (2) is used to lookup the magnitude of the oxygen gradient from premixed flamelet tables. In total, \(10^4\) freely propagating flames were computed between \(0<Z<0.5\) to create the look-up tables for computing \(D_O\) (see Table 1). The largest difference between \(\xi _{{\mathrm {Fiorina}}}(c_1)\) and \(\xi ({{\mathrm {CH}}}_4,{\mathrm {O}}_2)\) occurs downstream, where \(\xi _{{\mathrm {Fiorina}}}(c_1)\) predicts a larger portion of the flame to be non-premixed.

Figure 5g shows the regime marker based on the mass fraction of CO instead of \(\hbox {CH}_4\), as proposed by Som and Aggarwal (2010) (see Table 1 and the discussion in Sect. 1 about using CO instead of \(\hbox {CH}_4\)). According to this marker, the part of the flame near the nozzle, where the reaction zone is thin, consists of an inner non-premixed zone and an outer premixed zone, in contrast to \(\xi ({{\mathrm {CH}}}_4,{\mathrm {O}}_2)\), which predicts a purely premixed zone. Also, the premixed regions downstream predicted by \(\xi ({\mathrm {CO}},{\mathrm {O}}_2)\) are much thinner than those predicted by \(\xi ({{\mathrm {CH}}}_4,{\mathrm {O}}_2)\).

Fig. 6
figure 6

Instantaneous cutting planes at axial positions \(x/D=5\) (top) and \(x/D=20\) (bottom). In the left column, the chemical mode is depicted and on the right the normalized flame index in regions where \(\dot{Q}>10^7\) \(\hbox {W/m}^3\). The yellow line shows the iso-contour of \(Z=Z_{{\mathrm {st}}}\) and the green line the iso-contour of \(\dot{Q}=10^8\) \(\hbox {W/m}^3\). F denotes the fuel species, here \(\hbox {CH}_4\)

In Fig. 1 in Sect. 3, it was already pointed out that the flame shows mostly premixed characteristics near the nozzle and both premixed and non-premixed characteristics further downstream. Also, the reaction zones become broader with increasing axial position. This is illustrated in Fig. 6: In the top row, two instantaneous slices at \(x/D=5\) depicting the chemical mode (left) and flame index \(\xi (F,{\mathrm {O}}_2)\), where the fuel species F is \(\hbox {CH}_4\) (right), illustrate the flame. The right column is additionally conditioned to the heat release rate, showing only regions where \(\dot{Q}>10^7\) \(\hbox {W/m}^3\). The yellow line is the iso-contour of \(Z=Z_{{\mathrm {st}}}\), which shows that the reaction zone lies fully in the rich part of the flame. The green line is the iso-contour of \(\dot{Q}=10^8\) \(\hbox {W/m}^3\). This region of high heat release rate is located exactly where the \({{ CM }}\) has its zero-crossing, jumping from the explosive mode (positive values, red area) to negative values (blue area). High heat release rate near a zero-crossing of \({ CM }\) signifies a premixed reaction zone (Sect. 1) or auto-ignition. Likewise, the flame index on the right shows that the whole region of high heat release rate encompassed by the green iso-contour is premixed (red). Further downstream at \(x/D=20\) (bottom row of Fig. 6), the region of high heat release rate lies directly at \(Z=Z_{{\mathrm {st}}}\) with negative values of \({ CM }\), marking a non-premixed reaction zone. The region with moderate heat release rates \(\dot{Q}>10^7\) \(\hbox {W/m}^3\) is much broader (bottom right of Fig. 6, non-gray area) and contains both premixed and non-premixed parts as identified by the flame index.

4.2 Premixed-Dominated Regime

In this and the following subsections, a direct comparison of the different flame regime markers from Table 1 along instantaneous flame fronts will be given and different aspects of their applicability and behavior highlighted. As shown in Sect. 3, the flame is predominantly premixed at \(x/D<10\). Figure 7 shows the different flame markers along an instantaneous radial line at \(x/D=1\). All gradients and quantities are evaluated in the full 3D space and then plotted along that line. In Fig. 7a, a single heat release rate peak is visible, which lies at the location of the zero-crossing of CM (red line). This heat release rate peak is identified by the GFRI approach as premixed, because it lies within 100 \(\mu \hbox {m}\) of the zero-crossing. Using larger radii can lead to a wrong characterization, because the flame fronts are thin upstream and zero-crossings far away from the heat release rate peaks have to be excluded (see also the “Appendix” for a discussion on this). Further downstream, however, the flame becomes broader and this value has to be increased to 175 \(\mu \hbox {m}\) in order to correctly identify the flame regimes. These values were chosen empirically, so that zero-crossings are assigned to the correct heat release rate peaks. Premixed heat release rate peaks as classified by GFRI are marked by a red dot (from here on, red meaning premixed and blue non-premixed), leading to a value of \(\xi _{{\mathrm {GFRI}}}=1\) (or 100 %).

Fig. 7
figure 7

Direct comparison of flame regime markers along an instantaneous radial line at \(x/D=1\). Subfigures show: a profiles of \({ CM }\), \(\dot{Q}\) and Z. b values of different regime markers along the line

Figure 7b evaluates the other flame markers from Table 1 along that line. The profile of the heat release rate is overlaid to give a reference where a position is relative to the reaction zone. The percentage numbers on the right show the relative importance of combustion regimes in terms of heat release rate, computed from:

$$\begin{aligned} \bar{\xi }=\frac{\int \xi \,\dot{Q}\,{\mathrm {d}}r}{\int \dot{Q}\,{\mathrm {d}}r} \end{aligned}$$

Therefore, a value of 1 or 100 % means a fully premixed flame.

All regime markers classify this flame as mostly premixed. The only two exceptions are the flame index based on the gradient alignment of CO and \(\hbox {O}_2\), which classifies the inner part of the flame as non-premixed, and \(\xi _{{\mathrm {Drift}}}({\mathrm {CO}})\), which predicts a flame that has premixed character on the rich, inner side and mixed-mode character on the lean side. This result is consistent with findings of Wu and Ihme (2016) and shows that even though this flame is predominantly premixed, the evolution of the CO mass fraction is best described by a blending of premixed and non-premixed manifolds due to the flame being stretched and subjected to transient effects.

Fig. 8
figure 8

Direct comparison of flame regime markers along an instantaneous radial line at \(x/D=1\) (left) and \(x/D=3\) (right). Subfigures show: a and c: profiles of \({ CM }\), \(\dot{Q}\) and Z. b and d values of different regime markers from Table 1 along the line

In Fig. 8 on the left, the evaluation of the markers is done for a different time instance at the same radial line as Fig. 7 at \(x/D=1\). Although the flame is predominantly premixed at this position, other combustion modes appear as well. The heat release rate has two distinct peaks (Fig.  8a), where the highest heat release rate is directly located at the zero-crossing of \({ CM }\) and is therefore designated \(\dot{Q}^{{\mathrm {PF}}}\) by GFRI. The second heat release rate peak occurs at a position where \({ CM }\) is negative. However, since \(Z< Z_{{\mathrm {st}}}\), it is identified as part of a premixed flame. \(\dot{Q}^{{\mathrm {DF}}}\) in this case is zero, because within the region of interest (RoI), which is set in this work to \(\pm 0.5\) mm around the zero-crossing due to the broadening of the reaction zone, no heat is released in regions with \({ CM }<0\) and \(0.055>Z> 0.07\) (see the “Appendix” for more information). The flame is therefore classified as premixed by GFRI, which is consistent with most other markers in Fig. 8b. The only marker that identifies the outer part of the flame as non-premixed is \(\xi _{{\mathrm {Drift}}}\) in the last row of Fig. 8b. It is interesting to note that this second heat release rate peak is accompanied by a jump in \({ CM }\) entirely in the negative range.

Another situation where the different flame regime markers disagree is depicted in Fig. 8 on the right at the axial position \(x/D=3\). The maximum heat release rate value is located near the CM zero-crossing and is therefore identified as \(\dot{Q}^{{\mathrm {PF}}}\) by GFRI. Due to the broadening of the reacting zone, \(\dot{Q}^{{\mathrm {DF}}}\) is located on the outer side of the reaction zone, leading to the classification of a predominantly premixed flame. Similarly, the flame index based on \(\hbox {CH}_4\) and \(\hbox {O}_2\) gradients as well as its multi-component extension (second row from the bottom in Fig. 8d) characterize the flame as predominantly premixed. The flame index by Fiorina et al. and the marker based on drift term analysis predict a partially premixed flame, where the heat is released approximately equally in both combustion regimes.

4.3 Partially Premixed Regime

Fig. 9
figure 9

Direct comparison of flame regime markers along an instantaneous radial line at \(x/D=15\). Subfigures show: a profiles of \({ CM }\), \(\dot{Q}\) and Z. b Values of different regime markers from Table 1 along the line

For \(x/D>10\), the statistical average from Fig. 6 shows that heat is generated in equal parts from premixed and non-premixed regions. An instantaneous flame front in this region is depicted in Fig. 9. The heat release rate profile has three distinct peaks, where the global maximum heat release rate at \(r=5\) mm is identified as \(\max (\dot{Q}^{{\mathrm {PF}}})\) (red circle) and the middle peak is the maximum heat release rate \(\max (\dot{Q}^{{\mathrm {DF}}})\) (blue circle) outside the zero-crossing and within the RoI with \({ CM }<0\) and \(Z>Z_{{\mathrm {st}}}\), yielding \(\xi _{{\mathrm {GFRI}}}=0.21\). Most flame markers from subfigure b identify the inner part of the flame as non-premixed as well. The exceptions are the one by Fiorina et al., especially when using \(Y_{c_2}\) from Eq. (3) to retrieve the oxygen mass fraction gradient from the premixed tables, the flame index based on CO and the regime marker based on drift term analysis. The outer part of the flame (\(r>8\) mm) is identified by all regime markers as non-premixed. The flame index based on \(\hbox {CH}_4\) and \(\hbox {O}_2\) as well as the ones by Fiorina et al. (first three rows) identify a section of the flame at \(r=10\) mm as premixed. However, this is only an artifact of the numerical evaluation. The gradient of \(\hbox {CH}_4\) at that position is close to zero, so that the flame index becomes meaningless. Using instead the multi-species marker (second row from the bottom) extends the range of applicability of this marker and consistently identifies the outer part of the flame as non-premixed.

5 Summary and Conclusion

In this work, three types of fundamentally different flame regime markers from the literature have been applied to a simulation database of the compositionally inhomogeneous, partially premixed turbulent Sydney/Sandia flame in configuration FJ-5GP-Lr75-57, where the flame is fully resolved. The simulation has been performed using finite rate chemistry as well as detailed molecular diffusion. This allows computing the markers without the influence of combustion models as a post-processing step. When evaluating the combustion regime as a statistical average in terms of the heat generation, the upstream region of the flame (\(x/D<10\)) is predominantly premixed (lean premixed for \(x/D<2\) and rich premixed for \(2<x/D<10\)), with up to 90 % of heat being generated in regions with premixed characteristics. Downstream of that, heat generated from both the premixed and non-premixed regime becomes approximately equal.

The partially premixed character of this flame at \(x/D>10\) is not only created by having alternating premixed and non-premixed flamelets appear at the same location at different times, but also through direct interaction of premixed and non-premixed flame fronts as identified by all different regime markers. Even at \(x/D<10\), where the heat is generated in predominantly premixed regions, instantaneous flame fronts can show mixed characteristics. Even though direct comparison of the different types of regime markers shows discrepancies and further research for consistent regime identification is required, the following results can be summarized:

  1. 1.

    The flame index based on the alignment of methane and oxygen mass fraction gradients gives wrong results in parts of the reaction zone, where methane is fully depleted. Likewise, using CO instead of CH\(_4\) can only be used in the outer part of the flame to give consistent results. Therefore, a multi-species approach developed in the past (Lignell et al. 2011; Jangi et al. 2015; Minamoto et al. 2015; Ma et al. 2016; Pei et al. 2016; Wei et al. 2018; Pang et al. 2018; Tufano et al. 2018; Hu and Kurose 2019b; Wan et al. 2019; Wen et al. 2019; Akaotsu et al. 2020), where the fuel species is the combination of e.g. \(\hbox {CH}_4\), CO, \(\hbox {H}_2\) and \(\hbox {C}_2\hbox {H}_2\), is able to extend the range of applicability to the whole flame front in all investigated cases.

  2. 2.

    In almost all cases, results of the GFRI method \(\xi _{{\mathrm {GFRI}}}\) are consistent with the flame index based on a combination of fuel and intermediate species \(\xi ({{\mathrm {Multi}}},{\mathrm {O}}_2)\). The regime marker by Fiorina et al. in its original formulation has the same limitations as the classical Takeno flame index, i.e. the depletion of the fuel within the flame front, and is the only marker that predicts non-premixed characteristics at the inner part of the flame for \(x/D<10\) in some cases. Also, the exact way of tabulation for the premixed flamelets for comparing the oxygen gradient, e.g. the choice of progress variable, can have a strong impact on the prediction of \(\xi _{{\mathrm {Fiorina}}}\).

  3. 3.

    The Bilger mixture fraction Z and progress variable c are statistically strongly correlated. Although instantaneous flame fronts have mixed mode characteristics, the alignment of Z and c gradients are mostly parallel and anti-parallel.

  4. 4.

    The GFRI approach yields consistent results with the flame indices based on gradient alignment. For this, the radius around the CM zero-crossing used to find \(\dot{Q}^{{\mathrm {PF}}}\) has to be increased from 100 \(\mu \hbox {m}\) to 175 \(\mu \hbox {m}\) downstream where the flame becomes generally thicker.

  5. 5.

    The regime marker based on drift term analysis is species specific. As intermediate species have higher sensitivity to the choice of combustion model than main species, the drift terms for CO are evaluated. Even for instantaneous flame fronts, where all other regime markers predict a dominant regime in terms of premixed or non-premixed, \(\xi _{{\mathrm {Drift}}}({\mathrm {CO}})\) tends to predict a mixed mode. The reason is that \(\xi _{{\mathrm {Drift}}}({\mathrm {CO}})\) does not specifically determine the combustion regime but finds the optimal blending of two combustion models to represent the profile of the chosen species. In the presence of flame stretch, a blending of the premixed and non-premixed model improves the accuracy of the CO profile even though the flame shows otherwise dominantly premixed characteristics. It is also the only marker that identifies outer heat release rate peaks (in case of instantaneous flame fronts with two heat release rate peaks) near the nozzle as non-premixed. These peaks lie in the negative \({ CM }\) range.