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 flows are presented. Second, the basic parameters for characterizing turbulent diffusion flames are presented. Third and final, the classification techniques of local combustion regimes are described along with their phenomenological interpretation.
2.1 Turbulence
Turbulent flows 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 \(\text{q}\), the most immediate indicator is its Reynolds i.e. time average, which is defined as:
$$\overline{\text{q}}=\frac{1}{{\tau }}{\int }_{0}^{{\tau }}\text{q}\left(\text{t}\right)\text{d}\text{t}$$
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In variable-density flows, such as turbulent flames, it is convenient to introduce density weighting to account for the specific volume variations. The Favre-average [16] arises from this consideration:
$$\tilde{\text{q}}=\frac{1}{{\tau }\overline{{\rho }}}{\int }_{0}^{{\tau }}\text{q}\left(\text{t}\right){\rho }\left(\text{t}\right)\text{d}\text{t}$$
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Averaged values can be taken as a reference to address instantaneous values as fluctuations. Hence the Reynolds fluctuations are defined as \({\text{q}}^{{\prime }}\left(\text{t}\right)=\text{q}\left(\text{t}\right)-\overline{\text{q}}\) and the Favre fluctuations as \({\text{q}}^{{\prime }{\prime }}=\text{q}\left(\text{t}\right)-\tilde{\text{q}}\). Higher-order statistics are used to quantify the statistics of these fluctuations. The most relevant indicator is the turbulent kinetic energy, which is essentially the variance of the velocity fluctuations:
$$\tilde{\text{k}}=\frac{\overline{{\rho }{\text{u}}_{\text{i}}^{{\prime }{\prime }}{\text{u}}_{\text{i}}^{{\prime }{\prime }}}}{\overline{{\rho }}}$$
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The turbulent kinetic energy is a fundamental manifestation of turbulence as it indicates the magnitude of the flow unsteadiness and its influence 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 flamelet model [19] contain implicit or explicit dependencies on the turbulent kinetic energy [20].
In turbulent premixed flames, turbulent kinetic energy enhances the burning rate, although the flame speed sensitivity to \(\tilde{\text{k}}\) 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 significantly larger than the flame front, their effect on flame 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 flame speeds. Nonetheless, previous research[25] suggested the possibility of quenching in premixed flames with high turbulent intensity, but there is no scientific consensus concerning this claim [26]. In diffusion flames, 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 significantly faster than the mixing process, it can disrupt the local diffusion flame structure producing local quenching [9].
2.2 Turbulent Diffusion flames
In a diffusion flame, 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 briefly described in this sub-section.
The mixture fraction \(\text{Z}\) is used to assess the local mixing conditions regardless of the combustion progress. For a chemical mixture consisting of \(\text{N}\) species, this parameter is determined as follows:
$$\text{Z}=\sum _{\text{k}=1}^{\text{N}}{\text{Y}}_{\text{k}}{\text{Z}}_{\text{k}}$$
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Where \({\text{Y}}_{\text{k}}\) and \({\text{Z}}_{\text{k}}\) stand for the mass fraction and the mixture fraction of the \({\text{k}}^{\text{t}\text{h}}\) chemical species, respectively. The mixture fraction of a molecule is its hydrocarbon content mass fraction. For a generic chemical species with the formulation \({\text{C}}_{{\text{n}}_{\text{C}}}{\text{H}}_{{\text{n}}_{\text{H}}}{\text{O}}_{{\text{n}}_{\text{O}}}\) this parameter is determined as follows:
$${\text{Z}}_{\text{k}}=\frac{{\text{A}}_{\text{C}}{\text{n}}_{\text{C}}+{\text{A}}_{\text{H}}{\text{n}}_{\text{H}}}{{\text{A}}_{\text{C}}{\text{n}}_{\text{C}}+{\text{A}}_{\text{H}}{\text{n}}_{\text{H}}+{\text{A}}_{\text{O}}{\text{n}}_{\text{O}}}\approx \frac{12{\text{n}}_{\text{C}}+{\text{n}}_{\text{H}}}{12{\text{n}}_{\text{C}}+{\text{n}}_{\text{H}}+16{\text{n}}_{\text{O}}}$$
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Where \({\text{A}}_{\text{C}}\), \({\text{A}}_{\text{H}}\) and \({\text{A}}_{\text{O}}\) stand for the atomic mass weights of carbon, hydrogen, and oxygen. As can be seen, the mixture fraction is zero when only the oxidizer is present, and it tends to unity when the mixture consists only of the fuel. Near injection, the mixture fraction profile presents sharp gradients indicating the mixing of clusters with \(\text{Z}\approx 1\) and \(\text{Z}\approx 0\). At downstream positions, the \(\text{Z}\) field is smoothed, tending to the global mixture fraction \({\text{Z}}_{\text{f}\text{l}\text{a}\text{m}\text{e}}={\dot{\text{m}}}_{\text{C}\text{H}4}/({\dot{\text{m}}}_{\text{C}{\text{H}}_{4}}+{\dot{\text{m}}}_{{\text{O}}_{2}})\).
Besides the mixing of propellants, it is also necessary to evaluate the local development of chemical reactions. The progress variable is used for this purpose. For non-premixed combustion, the progress variable can be determined as suggested by Bray et al. [28]:
$$\text{c}=\frac{{\text{Y}}_{\text{c}}}{{\text{Y}}_{\text{c},\text{e}\text{q}}\left(\text{Z}\right)}$$
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Where \({Y}_{c}\) is the reaction progress variable, and \({\text{Y}}_{\text{c},\text{e}\text{q}}\left(\text{Z}\right)\) 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 definitions 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 significant proliferation of free radicals and hydroxyl molecules. For this reason, the mixture fraction definition used was the same in recent works [11, 30]:
$${\text{Y}}_{\text{c}}=\frac{{\text{Y}}_{{\text{C}\text{O}}_{2}}}{{\text{M}}_{\text{C}{\text{O}}_{2}}}+\frac{{\text{Y}}_{\text{C}\text{O}}}{{\text{M}}_{\text{C}\text{O}}}+\frac{{\text{Y}}_{{\text{H}}_{2}\text{O}}}{{\text{M}}_{{\text{H}}_{2}\text{O}}}+\frac{{\text{Y}}_{\text{O}}}{{\text{M}}_{\text{O}}}+\frac{{\text{Y}}_{\text{H}}}{{\text{M}}_{\text{H}}}+\frac{{\text{Y}}_{\text{O}\text{H}}}{{\text{M}}_{\text{O}\text{H}}}$$
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Where \({\text{M}}_{\text{l}}\) denotes the molecular weight of the \({\text{l}}^{\text{t}\text{h}}\) 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 defined as:
$${\chi }=2{\text{D}}_{\text{m}}{\left(\frac{\partial \text{Z}}{\partial {\text{x}}_{\text{i}}}\right)}^{2}$$
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Where \({\text{D}}_{\text{m}}\) 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:
$$\tilde{\chi }\propto \frac{\tilde{{\epsilon }}}{\tilde{\text{k}}}\tilde{{\text{Z}}^{{\prime }{\prime }2}}=\frac{\sqrt{\tilde{\text{k}}}}{\tilde{\text{L}}}\tilde{{\text{Z}}^{{\prime }{\prime }2}}$$
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Where \(\tilde{L}={\tilde{k}}^{\frac{3}{2}}/\tilde{\epsilon }\) denotes the length scale of the large eddies. In stable non-premixed flamelets fast chemistry, the heat release rate scales with the scalar dissipation rate at stoichiometric conditions [27] i.e. \(\dot{\text{q}}\propto {{\chi }}_{st}\). Following (9), it is clear that the local mixing intensity and burning rate are primarily driven by the turbulent kinetic energy.
In anchored flames with a stable recirculation zone, a standing flame 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 flames 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 flow rate at the inlet to account for this fact. One possible normalization is as follows:
$$\frac{1}{{{\delta }}_{\text{q}}}=\frac{\dot{\text{q}}}{{\dot{\text{V}}}_{\text{i}\text{n}\text{j}}{\text{Z}}_{\text{f}\text{l}\text{a}\text{m}\text{e}}\text{E}}$$
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Where \(\text{E}\) is the specific energy content of methane (\(\text{E}\approx 55.5 \text{M}\text{J}/\text{k}\text{g}\)) and \(\dot{\text{V}}={\rho }{\text{u}}_{\text{i}\text{n}\text{j}}\) the volumetric flow 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 flame thickness at stoichiometric conditions:
$$\frac{{{\delta }}_{\text{L}0}}{{{\delta }}_{\text{q}}}=\frac{\dot{\text{q}}{{\delta }}_{\text{L}0}}{\dot{\text{V}}{\text{Z}}_{\text{f}\text{l}\text{a}\text{m}\text{e}}\text{E}}$$
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The most important point of this normalization is the correction with respect to the incoming velocity, which allows comparing flows with different injection mass flow rates. More sophisticated normalizations might be possible, but the one presented in this paper suffices within the present work’s scope.
2.3. Flame classification
For stability and performance purposes, liquid rocket engines typically operate in non-premixed conditions with fuel excess. Despite this global configuration, 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 fields. The presented methods will be applied to the output of the performed numerical simulations to investigate the occurrence of the different burning conditions.
2.3.1. 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:
$$\text{C}{\text{H}}_{4}+2{\text{O}}_{2}\to 2{\text{H}}_{2}\text{O}+\text{C}{\text{O}}_{2}$$
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Which yields a stoichiometric mass O/F ratio of approximately 4, corresponding to \({Z}_{st}\approx 0.2\). 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 specific impulse. In addition, the high temperatures significantly promote carbon monoxide generation. As a consequence, the equilibrium mass fraction of carbon monoxide significantly outweighs that of carbon dioxide. In an extreme case, the corresponding adjusted global reaction can be written as:
$$\text{C}{\text{H}}_{4}+\frac{3}{2}{\text{O}}_{2}\to 2{\text{H}}_{2}\text{O}+\text{C}\text{O}$$
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Which implies a stoichiometric O/F of roughly 3, corresponding to \({\text{Z}}_{\text{s}\text{t}}\approx 0.25\). Besides the significant 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 flame (\({\text{T}}_{\text{a}\text{d}}>3400 \text{K}\)). Consequently, significant mass fractions of free radicals and hydroxyl are present at equilibrium. All these substantial deviations from the assumption in (12) demand the redefinition 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 \({\text{Z}}_{\text{s}\text{t}}\approx 0.2\). More specifically, the maximum value is found at \(\text{Z}\approx 0.25\). This is in good agreement with the previous reasoning since \({\text{Y}}_{\text{C}\text{O},\text{e}\text{q}}/{\text{Y}}_{{\text{C}\text{O}}_{2},\text{e}\text{q}}\approx 4\). Hence it can be expected that the redefined 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 \(\text{O}/\text{F}\approx 3\) 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 \({\text{Z}}_{\text{s}\text{t}}\approx 0.25\). It is important to remark that this specific definition is assumed to address the decoupling between local maximum heat release and standard stoichiometric oxidizer to fuel ratio. From a global standpoint, the classical definition should be considered.
2.3.2 Flame Index
In burning devices that operate in non-premixed conditions, the reacting flow simultaneously displays both premixed and non-premixed configurations 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 flame. Regime markers can be used to identify the local burning regime. The most commonly used marker is the flame index \(\text{F}\text{I}\) as defined by Yamashita et al. [36]:
$$\text{F}\text{I}=\frac{1}{2}+\frac{\nabla {\text{Y}}_{{\text{C}\text{H}}_{4}}\cdot \nabla {\text{Y}}_{{\text{O}}_{2}} }{2\left|\nabla {\text{Y}}_{{\text{C}\text{H}}_{4}}\right|\left|\nabla {\text{Y}}_{{\text{O}}_{2}}\right|}$$
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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 counterflow non-premixed configuration. There are several available definitions for a flame 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 definitions yield similar classifications overall. In the frame of the present work the definition of Yamashita et al. was the one considered due to its simplicity and extended usage.
An example of the different classifiers presented in this section can be visualized in Fig. 2. In this figure, an instantaneous detail of the shear layer for one of the simulated flames is displayed. This graph shows the flame 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 counterflow structure of a diffusion flame. Heat release is concentrated nearby stoichiometric conditions, where the local O/F maximizes the reacting capabilities of the flow. 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 specific to space propulsion applications. Due to the absence of nitrogen, the flammability limits are significantly widened, enabling combustion in extremely fuel and/or oxygen-rich conditions. Direct numerical simulations enable accessibility to every relevant physical field. Therefore, they offer the possibility to study the occurrence of these events and how they influence the combustion development process.