Combustion regimes in turbulent non-premixed �ames for space propulsion

Direct numerical simulations of non-premixed fuel-rich methane-oxygen �ames at 20 bar are conducted to investigate the turbulent mixing burning of gaseous propellants in rocket engines. The reacting �ow is simulated using the EBI-DNS solver, within the OpenFOAM frame. The transport of species is resolved with �nite rate chemistry, using a complex skeletal mechanism that entails 21 species. Two different �ames at low and high Reynolds numbers are considered to study the sensitivity of the �ame dynamics to turbulence. Regime markers are used to measure the probability of the �ow to burning in premixed and non-premixed conditions at different regions. The local heat release statistics are studied to understand the drivers in the development of the turbulent diffusion �ame. Despite the eminent non-premixed con�guration, a signi�cant amount of combustion takes place in premixed conditions. Premixed combustion is viable in both lean and fuel-rich regions, relatively far from the stoichiometric line. It is found that a growing turbulent kinetic energy is detrimental to combustion in fuel-rich premixed conditions. This is motivated by the disruption of the local premixed �ame front, which promotes fuel transport into the diffusion �ame. In addition, at downstream positions, higher turbulence enables the advection of methane into the lean core of the �ame, enhancing the burning rates in these regions. Hence, the primary effect of turbulence is to increase the fraction of propellants burnt in oxygen-rich and near stoichiometric conditions. As a consequence, the mixture fraction of the products shifts towards lean conditions, in�uencing combustion completion at downstream positions.


Introduction
Methane is a promising propellant for the new generation of space propulsion systems [1,2].
Nevertheless, critical aspects related to turbulent ignition, mixing and combustion challenge the development of new thrust chambers [3,4]. Computational Fluid Dynamics (CFD) software provides essential support in this process. Conventional methods applied therein, such as Reynolds-Averaged Navier-Stokes (RANS) solvers require turbulent combustion models to predict unclosed turbulent terms. This question has challenged the combustion community during the last century. Reacting ows escalate the inherent complexity of turbulence with the introduction of additional mechanisms of vorticity generation and dissipation [5,6]. Turbulence and combustion can promote or inhibit each other depending on the overlapping between their characteristic time and length scales [7,8,9]. Combustion in rocket engines is a particularly challenging case. Due to the high pressures and the absence of nitrogen the ow's reactivity is strongly accentuated, dwindling chemical length and timescales [10]. As a consequence, the Damköhler number is very high enabling the occurrence of exotic phenomena such as ame-generated turbulence [11,12] and intermittency between combustion in lean and fuel-rich conditions [13]. Knowledge of these phenomena is essential in the formulation of effective turbulent combustion models. Hence, it is necessary to investigate the conditions in which the combustion of injected propellants develops in a rocket combustor. The present text addresses this question using direct numerical simulations, representing turbulent mixing and combustion near the injection region of a methane-oxygen engine at high pressure. The resulting databases are post-processed to investigate the occurrence combustion in different mixing con gurations within this environment.
The rest of the paper is organized as follows. The next section provides a theoretical background, that provides necessary concepts concerning turbulence and non-premixed combustion. The third section describes the numerical simulation strategy. The numerical results are analyzed in the fourth section.
Finally, the fth section summarizes the concluding remarks of this research.

Theoretical Background
The present section introduces basic notions that are required to discuss the results of the conducted numerical simulations. The text is structured in three parts. First, basic concepts of turbulence for variable-density ows are presented. Second, the basic parameters for characterizing turbulent diffusion ames are presented. Third and nal, the classi cation techniques of local combustion regimes are described along with their phenomenological interpretation.

Turbulence
Turbulent ows are characterized by an unstable nature, which precludes reaching steady-state conditions. Due to their chaotic features [14,15], a statistical approach is deemed convenient. For a turbulent quantity , the most immediate indicator is its Reynolds i.e. time average, which is de ned as: 1 In variable-density ows, such as turbulent ames, it is convenient to introduce density weighting to account for the speci c volume variations. The Favre-average [16] arises from this consideration: 2 Averaged values can be taken as a reference to address instantaneous values as uctuations. Hence the Reynolds uctuations are de ned as and the Favre uctuations as .
Higher-order statistics are used to quantify the statistics of these uctuations. The most relevant indicator is the turbulent kinetic energy, which is essentially the variance of the velocity uctuations: The turbulent kinetic energy is a fundamental manifestation of turbulence as it indicates the magnitude of the ow unsteadiness and its in uence on the momentum conservation equations. Hence, it denotes the degree of forced convection which is essential in turbulent combustion and mixing processes. Indeed, turbulent combustion models such as the Eddy Dissipation Concept (EDC) [17,18] and the amelet model [19] contain implicit or explicit dependencies on the turbulent kinetic energy [20].
In turbulent premixed ames, turbulent kinetic energy enhances the burning rate, although the ame speed sensitivity to decreases with growing turbulence following the so-called bending effect [21]. This is caused by a change in the overlapping between chemical and turbulent scales [22,23,24]. If vortexes are signi cantly larger than the ame front, their effect on ame development is mainly an increase in the effective surface due to wrinkling. If the turbulent structures are smaller, the enhancing effects of turbulence primarily occur in terms of diffusion increase. In both cases, it is expected that greater turbulence yields higher ame speeds. Nonetheless, previous research [25] suggested the possibility of quenching in premixed ames with high turbulent intensity, but there is no scienti c consensus concerning this claim [26]. In diffusion ames, if the chemistry is very fast compared to turbulence, an enhancement of the burning rate with respect to chemistry takes place [27]. This is primarily motivated by the fact that the mixing intensity grows with the turbulent kinetic energy. However, if turbulence becomes signi cantly faster than the mixing process, it can disrupt the local diffusion ame structure producing local quenching [9].

Turbulent Diffusion ames
In a diffusion ame, oxidizer and fuel are simultaneously mixing and reacting. The development of these processes is measured with the mixture fraction and the progress variable, which are brie y described in this sub-section.
The mixture fraction is used to assess the local mixing conditions regardless of the combustion progress. For a chemical mixture consisting of species, this parameter is determined as follows: 4 Where and stand for the mass fraction and the mixture fraction of the chemical species, respectively. The mixture fraction of a molecule is its hydrocarbon content mass fraction. For a generic chemical species with the formulation this parameter is determined as follows: Where is the reaction progress variable, and its equilibrium composition for the present mixture fraction. The reaction progress variable is a scalar that indicates the development of the combustion process. For methane combustion, the de nitions devised by Pierce and Moin [29] are often used. However, in space propulsion applications, due to the absence of inert gases, high temperatures take place, with a signi cant proliferation of free radicals and hydroxyl molecules. For this reason, the mixture fraction de nition used was the same in recent works [11,30] :   7 Where denotes the molecular weight of the chemical species. If chemical reactions are very fast compared to turbulent processes, then the progress variable and the mixture fraction are algebraically related [31]. Therefore, the progress variable dynamics are driven by the mixing intensity. This physical phenomenon is evaluated with the scalar dissipation of the mixture fraction, which is de ned as:

8
Where is the mass diffusivity of the reacting mixture. The scalar dissipation rate is analogous to the dissipation rate of the turbulent kinetic energy. Indeed, closure models [19,32,33] relate them as: Where denotes the length scale of the large eddies. In stable non-premixed amelets fast chemistry, the heat release rate scales with the scalar dissipation rate at stoichiometric conditions [27] i.e.
. Following (9), it is clear that the local mixing intensity and burning rate are primarily driven by the turbulent kinetic energy.
In anchored ames with a stable recirculation zone, a standing ame can be maintained regardless of the injection velocity since continuous mixing is enabled near the injection posts. Hence, as the velocity increases, the heat release rate grows as well due to the greater rate of incoming fresh reactants. This fact can hinder the comparison between ames burning with different injection velocities. Such is the case for the numerical simulations performed in the present work. For this reason, the heat release rate should be normalized with the mean ow rate at the inlet to account for this fact. One possible normalization is as follows:

10
Where is the speci c energy content of methane ( ) and the volumetric ow rate at injection. This value has dimensions of 1/length, indicating the longitude required for the mixture to obtain all the available energy. Therefore, one suitable reference value to normalize it is the laminar ame thickness at stoichiometric conditions:

11
The most important point of this normalization is the correction with respect to the incoming velocity, which allows comparing ows with different injection mass ow rates. More sophisticated normalizations might be possible, but the one presented in this paper su ces within the present work's scope.

Flame classi cation
For stability and performance purposes, liquid rocket engines typically operate in non-premixed conditions with fuel excess. Despite this global con guration, at local and instantaneous levels, combustion can take place in premixed and lean conditions as well. The occurrence of these processes is dependent on the turbulent mixing of the propellants. Direct numerical simulations allow the detailed observation of these local phenomena. The present sub-section discusses objective criteria to quantify and classify the local burning regime based on the instantaneous physical elds. The presented methods will be applied to the output of the performed numerical simulations to investigate the occurrence of the different burning conditions.

Fuel-rich vs. lean
The classical procedure to calculate the stoichiometric ratio of a hydrocarbon is based on the assumption that the reactants are oxidized into a mixture entirely composed of water and carbon dioxide. For the case of methane, this corresponds to the following global reaction scheme:

12
Which yields a stoichiometric mass O/F ratio of approximately 4, corresponding to . In atmospheric combustion applications that operate in lean conditions, this approximation is quite accurate since the carbon monoxide content is very small compared to carbon dioxide. However, for combustion in space propulsion applications fuel-rich mixtures are desired to maximize the speci c impulse. In addition, the high temperatures signi cantly promote carbon monoxide generation. As a consequence, the equilibrium mass fraction of carbon monoxide signi cantly outweighs that of carbon dioxide. In an extreme case, the corresponding adjusted global reaction can be written as:

13
Which implies a stoichiometric O/F of roughly 3, corresponding to . Besides the signi cant portion of carbon monoxide, additional particularities of combustion in the frame of space propulsion further entangle the equilibrium composition. Due to the high pressures and the absence of nitrogen, very high temperatures are reached at the ame ( ). Consequently, signi cant mass fractions of free radicals and hydroxyl are present at equilibrium. All these substantial deviations from the assumption in (12) demand the rede nition of the stoichiometric condition. A reasonable criterion is to choose the O/F that maximizes the volumetric heat release. The integrated heat release rate was calculated as a function of the mixture fraction using Cantera [34] and a complex chemical mechanism [35]. The result is displayed in Fig. 1. As it can be seen, the mixture fraction that maximizes heat release takes place at values higher than the classical assumption of . More speci cally, the maximum value is found at . This is in good agreement with the previous reasoning since . Hence it can be expected that the rede ned stoichiometric value is closer to the one in the assumption (13). Besides the graph in Fig. 1, additional evidence for the better suitability of an as a stoichiometric value is provided during the discussion of the results.
The shift in the maximum heat release shift is relevant since most models for non-premixed combustion require the knowledge of properties at stoichiometric conditions. Within the present work, it will be assumed that the stoichiometric mixture fraction in high-pressure methane-oxygen combustion corresponds to . It is important to remark that this speci c de nition is assumed to address the decoupling between local maximum heat release and standard stoichiometric oxidizer to fuel ratio.
From a global standpoint, the classical de nition should be considered.

Flame Index
In burning devices that operate in non-premixed conditions, the reacting ow simultaneously displays both premixed and non-premixed con gurations at a local level. These variations are driven by the local mixing conditions, which can enable premixed combustion despite the predominant diffusive nature of the ame. Regime markers can be used to identify the local burning regime. The most commonly used marker is the ame index as de ned by Yamashita et al. [36]: 14 The original formulation has been re-normalized so that zero corresponds to non-premixed and unity to premixed. The relationship between this numerical value and the local combustion regime can be easily deduced by the physical meaning of the mathematical expression. If the gradients of fuel and oxidizer are aligned, then the local mixing conditions can be deemed premixed. Contrarily, opposed gradients of the reactants correspond to a counter ow non-premixed con guration. There are several available de nitions for a ame index proposed in recent works [37,38,39,40,41,42]. These markers have been extensively reviewed in a recent study [43], where it was found that most de nitions yield similar classi cations overall. In the frame of the present work the de nition of Yamashita et al. was the one considered due to its simplicity and extended usage.
An example of the different classi ers presented in this section can be visualized in Fig. 2. In this gure, an instantaneous detail of the shear layer for one of the simulated ames is displayed. This graph shows the ame index, mixture fraction, and temperature to illustrate the different regimes in which combustion can take place in space propulsion applications. As can be seen, the central region of the shear layer presents the classical counter ow structure of a diffusion ame. Heat release is concentrated nearby stoichiometric conditions, where the local O/F maximizes the reacting capabilities of the ow. However, in certain positions far away from the stoichiometric line, the gradients of methane and oxygen are aligned, denoting local premixed combustion. This circumstance is very speci c to space propulsion applications. Due to the absence of nitrogen, the ammability limits are signi cantly widened, enabling combustion in extremely fuel and/or oxygen-rich conditions. Direct numerical simulations enable accessibility to every relevant physical eld. Therefore, they offer the possibility to study the occurrence of these events and how they in uence the combustion development process.

Numerical Simulation Case Setup
The numerical simulations aim at representing the turbulent combustion process that occurs in modern rocket engines in locations near injection and far away from walls. The turbulent reacting ow is resolved with the compressible solver EBI-DNS [44,45,46,47], which is embedded in the open-source software OpenFOAM [48,49]. This code uses an implicit scheme to resolve the transient conservation equations for mass, momentum, energy, and species with the Finite Volume Method (FVM) [50,51]. EBI-DNS has been applied and validated in several combustion-related problems in recent years [52,53,54,55,56]. The Cantera [34] routines are used for the computation of detailed chemistry and transport properties, using the mixture-averaged transport model described by Kee et al. [57]. The reaction mechanism developed by Slavinskaya et al. [35] is used to determine the chemical source terms of methane combustion using the nite rates method. This skeletal mechanism consists of 21 species and 97 reactions, and it was devised for combustion in space propulsion applications high-pressure. This mechanism was applied in several numerical studies of combustion for scale rocket combustors [58,59]. Relevant data concerning combustion is provided in TABLE I.  . These methods are based on discretizing a reference turbulent kinetic energy spectrum in harmonics with random amplitude and phases. These waves are thereafter superposed to provide a synthetic velocity eld that complies with prescribed 2nd order turbulent statistics. The statistics were selected to resemble the output of RANS simulations for a scale methane rocket combustor discussed in a previous work [63]. The described simulation strategy is summarized in Fig. 3. Due to the high computational cost associated with DNS, the hydraulic diameter of the injector is one order of magnitude smaller than the actual setup. As a consequence, the global Reynolds number and the turbulent Reynolds number are smaller than the reference device. Despite this shift, most of the relevant turbulent and chemical features are preserved. Hence, the simulations can be assumed to capture the relevant aspects of turbulent mixing and combustion. . The ame front is resolved with varying resolution since the ame front thickness varies. The chosen grid size ensures a resolution of the instantaneous ame front realizations in the order of 10 cells. This is generally enough, although a higher resolution would be desirable. The convergence of the turbulent statistics for the used setup has been addressed in a prior work [30]. The effect of increasing the resolution is marginal. More speci cally, a grid size 2.5 times larger yields maximum errors in the order of 5% and axially averaged properties below 1%. A ner grid (halving the cell volume) leads to discrepancies in the axially averaged properties in the order of 0.5 %. Hece, it can be concluded that the resolution of the used mesh is high enough to allow the usage of the simulation results for turbulence research purposes. The timestep size is chosen in an adaptive fashion to ensure that the maximum acoustic CFL number is below 0.9. This corresponds to maximum convective CFL numbers in the order of 0.05 for and 0.25 for . Moreover, due to the implicit nature of the used solver, the ful llment of the general condition is not required to achieve numerical stability. Such low CFL numbers, coupled with the cubic spatial differentiation scheme, ensure the minimization of numerical dissipation, as discussed elsewhere [56]. This section is devoted to the discussion and analysis of the results obtained in the performed numerical simulations. The discussion is structured in three parts. The rst and the second sub-sections i.e., 4.1 and 4.2 discuss the overall results of simulations and respectively. The ame development is discussed from a phenomenological standpoint to identify the main events and the driving physical mechanisms. In the third sub-section (i.e., 4.3), the different interactions between turbulent mixing and combustion are discussed to investigate the sensitivity of the burning process with respect to the turbulent kinetic energy.

Simulation
The simulation is characterized by low inlet turbulence, with a ow behavior close to laminarity. An example of the instantaneous ame development can be observed in Fig. 4. In this illustration, the ame classi cation is displayed at the top, and both positive and negative heat release rate can be visualized at the center and bottom, respectively.
The ow structure near injection resembles a triple ame, with a non-premixed branch surrounded by lean and fuel-rich premixed branches. Negative heat release can be observed in the transition from fuel-rich premixed to fuel-rich non-premixed. This process is on account of the cracking reactions, which are slightly endothermic. The fuel-rich premixed branch displays a discontinuous axial development, with local breakups. This is caused by the forced advection of methane into the ame core. Due to its higher diffusivity, methane bene ts more than oxygen from the moderate turbulence. This different sensitivity to turbulence shifts the local O/F in the thick premixed regions outside the ammability limits. Contrarily, the lean premixed branch displays a more stable con guration. As expected, most of the heat release is obtained from the non-premixed amelets. Nevertheless, a signi cant amount of combustion occurs in the lean premixed side while both shear layers are still present. This trend is interrupted by , when both shear layers merge into a single structure. From this point, the combustion at the lean core signi cantly decreases, and the majority of chemical reactions develop on the non-premixed side. Figure 5 was elaborated to statistically summarize the observed processes. The right manifold can be used to investigate the probability of occurrence of the different classi cations. Most of the observations take place in non-premixed conditions ranging from moderately lean to signi cantly fuel rich. This is in good agreement with the example in Fig. 4, where it can be observed that the non-premixed amelet is predominant. There is a signi cant portion of data in premixed conditions with extremely lean and fuelrich mixtures. However, only the lean premixed regions deliver heat release rates in the same order of magnitude as the non-premixed amelets. Intermediate ame indexes are rather unusual. This is due to the quasi-bimodal distribution of the ame index which was observed in the examples of Fig. 2 and by other researchers [43]. These graphs provide valuable insights concerning the overall combustion process. However, it is important to remark on their strong dependency on geometry. For example, if the simulation domain were signi cantly longer, the statistics would shift toward the equilibrium conditions. This would imply a greater concentration of points towards , with a lower averaged heat release rate.

Simulation
An analogous analysis was conducted with the database of simulation . Unlike , this simulation presents a Reynolds number high enough to display fully turbulent features. An example of instantaneous ame classi cation and heat release rate can be visualized in Fig. 6.
As it can be seen, there is a greater disruption in the fuel-rich premixed region as this combustion regime only occurs in isolated regions. This is caused by the greater turbulent kinetic energy, which furthermore promotes the transport of methane into the ame core. Another remarkable fact is that the negative heat release due to cracking reactions near injection takes place intermittently. This effect is thanks to ame stretching, which can locally compensate for the effect of endothermic reactions if the curvature is high enough. This effect is discussed in greater detail in the following sub-section. Negative heat release can be observed at downstream positions as well. However, this result is mainly motivated by recombination reactions of combustion products.
Overall, the absolute heat release rate is substantially larger in compared to . This is mainly a consequence of the greater mass ow rate of propellants, which enhances the incoming rate of fresh reactants and the turbulent kinetic energy , which is proportional to the scalar dissipation rate . Nevertheless, the ow in simulation is able to keep up a signi cant burning rate in the ame core with learn premixed conditions following the merging of the shear layers. To complete this section, the 2D joint functions are displayed in Fig. 7. Little differences can be appreciated in the joint 2D probability density function. The most remarkable one is the greater prevalence of lean premixed con gurations. This is caused by the greater presence of this regime at downstream positions, as can be seen comparing Fig. 4 and Fig. 6. Concerning the normalized heat release, it can be seen that it is overall higher regardless of the considered con guration. However, enhancements are more remarkable in the lean regions. The differences in the heat release trends are discussed in greater detail in the following sub-section.

Comparison
Despite the remarkably different ow characteristics, for both ames, chemistry remains very fast compared to turbulence. To prove this claim, the progress variable as a function of the mixture fraction is represented in Fig. 8. As it can be seen, both ames follow the trend expected for fast chemistry [27,31].
Flame displays a slight deviation due to its higher turbulent kinetic energy, but the differences are overall negligible. The maximum value of the progress variable is obtained at , which further supports the discussed necessity to consider a shift in the stoichiometric value. The maximum variance of the progress variable conditional to is in the order of , corresponding to , for the rest of the domain, holds. Therefore, it can be assumed that combustion is limited by mixing since for a given mixture fraction the corresponding progress variable is reached almost immediately. This point is crucial for the interpretation of the results discussed in this section. Since the local heat release rate , which scales with , is essentially dependent on the mixture fraction dynamics.
Therefore, the in uence that turbulence exerts on the ame development occurs in terms of heat and mass transfer enhancements caused by forced convection.
To further study the in uence of turbulence in the mixing process, the probability distributions in Fig. 5 and Fig. 7 shall be compared. To assess the sensitivity of theses joint distributions with respect to inlet turbulence, the relative increase is displayed in Fig. 9. This graph provides a rough indication of the effects that a turbulence enhancement has with respect to mixing. As it can be seen, there is a greater amount of data toward the horizontal center. This is due to the improved mixing, which promotes the concentration of data with respect to the global mixture fraction i.e . However, the probability of presenting extreme conditions ( ) is enhanced as well. The motivation for this difference is the smaller residence time of the molecules in simulation . Since turbulence and mean velocity are increase simultaneously, the enhanced mixing is accompanied by a shorter stay time, which increases the possibility of an unmixed cluster reaching the outlet. These effects illustrate that although these comparative analyses provide valuable information, they should be approached carefully and critically.
To study the variations in the heat release, this value was averaged conditionally to the mixture fraction for both simulations. The result is displayed in Fig. 10. As it can be seen, provides a higher heat release overall, with the only exception being extremely lean regions.
Thanks to the fact that the and are algebraically related, most of the dependencies with the geometry and residence time are eliminated. Therefore, the differences in Fig. 10

Near Injection mixing
To approach the turbulent mixing dynamics near injection, a detail of the simulated ame is presented in Fig. 11. In this gure, the instantaneous stoichiometric line is presented along with the line marking the instantaneous border between premixed and non-premixed classi cation. Hence, the region between these two isolines corresponds to the lean side of the local diffusion ame. Overall, the ame exhibits a greater amount of normalized heat release. presents similar values in the stoichiometric line, but it only keeps up in the stretched regions due to the enhanced convection. Moreover, in these positions, the negative heat release due to cracking reactions can be suppressed thanks to the high local curvature, which fosters the heat transfer from the stoichiometric line to the fuel-rich zone. Indeed, it can be seen that the regions with non-negative heat release only appear where there is a signi cant stretching.

Downstream mixing
The ame development dynamics are dramatically altered once the merging of the shear layer takes place. To study the driving mechanisms, a detail of the local heat release for both simulations is presented in Fig. 12. In this illustration, the transparency is varied to identify the local burning regime. As can be seen, the lean side of the diffusion ame generates similar heat release in both simulations, although in simulation the observed values are slightly larger.
The main differences between both simulations take place at the ame core, where the mixture fraction is signi cantly below its stoichiometric value. In Simulation , intermittency between premixed and nonpremixed combustion can be observed. This result points to unstable burnability due to insu cient mixing intensity. This behavior contrasts with what is observed in , where the ame core burns mostly in premixed conditions. There, the heat release greatly varies in the two ames since it depends on the local strain and availability of reactants. In both cases, a heat release rate drop can be observed from the lean edge of the diffusion ame to the oxygen core. However, heat release near the diffusive region is signi cantly larger in , and the values decrease faster. This decreasing trend is motivated by the lower availability of burnable reactants, as the mixture fraction tends to . Although heat release at the core is smaller for simulation , the signi cantly higher burning rater close to the diffusive side enables a greater heat release in lean conditions overall. This process is behind the trend observed in Fig. 10 towards the lean side.
The differences in burning rate between both simulations are primarily driven by the different evolution of the scalar dissipation. To discuss the in uence of this variable, instantaneous 2D cuts of its value are displayed in Fig. 13.
As expected, the scalar dissipation rate is overall greater in . This is caused by the larger turbulent kinetic energy at the inlet. The most remarkable difference between both simulations corresponds to the behavior following the merging of the shear layers. In simulation , the mixing intensity decreases after the shear layers collapse. Contrarily, in , an increase in scalar dissipation follows the merging of the shear layers. This is motivated by the larger wrinkling of the shear layer due to the integrated effects of turbulence departing from injection. In , the shear layers are mostly aligned with the injection direction and, consequently, between themselves. Hence, their merging produces little additional shear, and global dissipation dominates. Nevertheless, in , turbulence has enough energy to create substantial oscillations of the diffusive ame structures prior to their merging. In such a scenario, the collapse becomes more abrupt, enhancing the local mixing intensity. This phenomenon enables the larger heat release rates as premixed combustion becomes viable at the core, as displayed in Fig. 12 for . If the scalar dissipation rate becomes too small, the possibility of combusting in a premixed in a stable way is disabled. This is evident in , but it can occur in localized regions in as well, as can be observed in Fig. 6  Non-premixed methane-oxygen turbulent diffusion ames were studied using direct numerical simulations with nite-rate chemistry. The main purpose of this research was to study the ame development in the frame of modern space propulsion systems. A novel de nition for the stoichiometric oxidizer-to-fuel ratio is proposed to enhance the analyses quality. The proposed de nition is better suited to reaction engines since the amount of carbon dioxide in equilibrium is small compared to carbon dioxide, altering the main assumption in which the classical de nition of stoichiometric conditions is grounded.
The rede ned stoichiometric value is used in conjunction with ame regime markers to classify the local combustion regime. Hence, two main degrees of freedom are considered: fuel-vs. oxygen-rich and premixed vs. non-premixed. The statistics of the heat release rate were analyzed conditional to these conditions with varying Reynolds numbers to investigate the in uence of turbulence in the ame development process. This investigation was performed by a comparative analysis of one mildly turbulent ame ( ) and a fully turbulent one ( ). For both ames, chemistry is very fast compared to turbulence. Hence, combustion progress is limited by mixing, and differences in the heat release dynamics can be primarily attributed to the forced convection variations. It is found that despite the eminent non-premixed con guration of the burning device, a signi cant amount of heat release occurs in lean premixed conditions. This process occurs primarily at the ame core following the merging of the shear layers. With growing turbulence, this merging becomes more abrupt due to the increased decorrelation between the two interacting diffusive ame structures.
The results presented here point out several challenges concerning combustion in systems that resort to oxygen as oxidizer. Due to the wider ammability limits, combustion in lean premixed conditions can become signi cant. This effect indicates the necessity to consider a combination of premixed and nonpremixed combustion models in the simulation in turbulent combustion simulations for space propulsion systems. In addition, the de nition of the stoichiometric conditions should be rede ned to account for the great carbon monoxide in equilibrium. Informed consent This research did not involve human participants Compliance with Ethical Standards

Declarations
Con ict of interests The authors declare that they have no con ict of interests Integral volumetric heat release as a function of the mixture fraction for methane-oxygen combustion at 20 bar