# Effects of Composition Heterogeneities on Flame Kernel Propagation: A DNS Study

^{*}

## Abstract

**:**

_{2}and nitrogen oxide NOx emissions of the transportation sector [1]. In 2020, the global annual international aviation emissions are already around 70% higher than in 2005. The International Civil Aviation Organization (ICAO) forecasts that, in the absence of additional measures, they could grow by over further 300% by 2050 [2]. Unfortunately, the current aeronautics propulsion systems have almost reached their technological limits and do not offer too many opportunities for a drastic reduction in greenhouse gas emissions. Thus, new propulsion concepts based on technological breakthroughs must be sought to improve the efficiency of the energy systems and meet the necessary emission targets, such as stabilizing CO

_{2}emissions at 2020 levels by requiring airlines to offset the increase of their emissions after 2020.

## 1. Governing Equations and Numerical Methods

## 2. Flow Configuration

## 3. DNS Database

#### Description of the Mixture Fraction and the Reaction Progress Variable

_{n}H

_{m}, the local equivalence ratio is expressed as follows

_{2}/H

_{2}O, the triplet CO/CO

_{2}/H

_{2}O and the couple CO/CO

_{2}.

_{2}/H

_{2}O) and (CO/CO

_{2}/H

_{2}O) respect criteria (iii) for all the isocontours except those in the vicinity of $c=1$ (on the side of the burnt gases), in this case $c=0.99$, which is relatively distant from the rest of the isocontours and not parallel to them. Indeed, the intermediate species and the radicals containing hydrogen continue to react in the burnt gases. Thus, these two definitions are also rejected. Finally, the introduction of the definition based on the couple CO/CO

_{2}, the isocontour $c=0.99$ approaches the others and the criterion of “parallelism” is respected. In addition, it combines a good discretization on the mesh with a monotony between 0 and 1. This definition could therefore be considered, since it respects the three constraints mentioned earlier. The transport equation of c can be written as (see Appendix A for a detailed development of this equation)

## 4. Results and Discussion

#### 4.1. Preferential Propagation

#### 4.1.1. Influence on Flame Surface Generation Mechanisms

#### 4.1.2. Flame Surface Density Budget

#### 4.2. Effect of Stratification on Displacement Flame Speed

_{2}. Therefore, the presence of heterogeneities induces a decrease in the reactive component of the speed of displacement ${S}_{d}$. We reiterate that this effect is more pronounced in large ${L}_{\Phi}$ as can be seen from the results depicted in the Figure 24.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Derivation of the Transport Equation from the Progress Variable

## References

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**Figure 1.**Turbulent combustion diagram in presence of $\Phi $ heterogeneities within moderate characteristic length and segregation rate. The green dot corresponds to the average equivalence ratio.

**Figure 2.**Diagram of the process of quadrants swapping to represent the deposit of the flame kernel at different places in a turbulent field.

**Figure 3.**Illustration of the process of swapping the quadrants of a speed field. Superposition of the contours of the vorticity fields and the generated speed fields.

**Figure 4.**Indexing of heterogeneous turbulent calculations. Fields of heat release superimposed on the local equivalence ratio fields for the eight computations corresponding to the case BH at $t/{\tau}_{F}=5$.

**Figure 6.**Isocontours of the progress variable for the BH at $t=8{\tau}_{F}$ computed with definitions based on (

**a**) fuel consumption Equation (20), (

**b**) the couple H

_{2}/H

_{2}O, (

**c**) the triplet CO/CO

_{2}/H

_{2}O and (

**d**) the couple CO/CO

_{2}.

**Figure 7.**Probability density function (PDF) of the mixture fraction on the side of fresh side, which corresponds to the isocontour $c=0.1$, at three instants $0.5$, $2.5$ and $5{\tau}_{F}$.

**Figure 9.**PDF of the alignment angle between the gradients of the progress variable and the mixture fraction on the flame front.

**Figure 10.**PDF of normalized curvature evaluated at the flame front at normalized instant $t=5{\tau}_{F}$.

**Figure 13.**Joint PDF of the displacement speed and the rate of tangential stretching on the flame front at $t=5{\tau}_{F}$.

**Figure 15.**Terms of the transport equation of the flame surface density function $\parallel \mathbf{\nabla}c\parallel $ as a function of the progress variable for the homogeneous case at $t=5{\tau}_{F}$. The difference between the right and left hand side of the Equation (29) is given by black points. (

**a**): scattering of the point cloud, (

**b**): averages conditioned by c.

**Figure 16.**Conditioned averages of the terms (

**a**)${\mathcal{T}}_{\sigma ,1}$, (

**b**) ${\mathcal{T}}_{\sigma ,2}$ and (

**c**) ${\mathcal{T}}_{\sigma ,3}$ of the flame surface density function budget as a function of the progress variable at $t=5{\tau}_{F}$.

**Figure 17.**Impact of composition heterogeneities on the displacement speed of the flame front (iso-c = 0.1). Left (

**a**): PDF at $t=5{\tau}_{F}$, right (

**b**): temporal evolution.

**Figure 18.**Terms of the progress variable c budget obtained for the case BH at $t=5{\tau}_{F}$. Left (

**a**): scatterplot, middle (

**b**): averages conditioned by c, right (

**c**): residual.

**Figure 19.**Sum of the mass fractions of CO and CO

_{2}at equilibrium as a function of the mixing fraction for the case BH at $t=5{\tau}_{F}$.

**Figure 20.**Influence of the unitary Lewis number hypothesis. Left (

**a**): Evolution of the Lewis number of the fuel, of CO and of CO

_{2}as a function of the progress variable for the case BH at $t=5{\tau}_{F}$, right (

**b**): profiles of the laminar flame velocity and thickness in the considered ranges of variations.

**Figure 21.**Evolution of the components of the displacement speed as a function of the progress variable for the case BH at $t=5{\tau}_{F}$.

**Figure 22.**Evolution of the PDF of the normal component of ${S}_{d}$ for three levels of the progress variable at $t=5{\tau}_{F}$.

**Figure 23.**(

**a**) Illustration of the contributions relative to the variations in surface, density and diffusion, in the normal displacement speed, for the case BH at $t=5{\tau}_{F}$. (

**b**) PDF of the contribution related to the variation of the flame surface density in the normal displacement speed on the isocontour c = 0.7.

**Figure 24.**Evolution of the PDF of the reactive component of the displacement speed for three levels of the progress variable at $t=5{\tau}_{F}$.

**Figure 25.**Evolution of the PDF of the differential diffusion component of the displacement speed for three levels of the progress variable at $t=5{\tau}_{F}$.

**Figure 26.**Average relative deviations of the normal (

**left**), reactive (

**right**) and differential diffusion (

**middle**) components from their homogeneous counterparts at $t=5{\tau}_{F}$.

**Table 1.**Characteristics of the direct numerical simulation (DNS) database simulations. Turbulence parameters and dimensionless numbers (Da, Ka and Re${}_{t}$) are given for the initial fields.

Operating conditions | ${T}_{u}\phantom{\rule{3.33333pt}{0ex}}$(K) | 700 |

$\overline{\Phi}$ | 0.8 | |

P (bar) | 5.0 | |

Turbulence | ${l}_{t}\phantom{\rule{3.33333pt}{0ex}}$(mm) | 0.4 |

${u}_{\mathrm{rms}}$ (m.s${}^{-1}$) | 1.0 | |

$R{e}_{t}$ | 29 | |

${\tau}_{t}\phantom{\rule{3.33333pt}{0ex}}$($\mathsf{\mu}$s) | 381 | |

Combustion | ${S}_{L}^{0}$ (m.s${}^{-1}$) | 0.98 |

${\delta}_{L}\phantom{\rule{3.33333pt}{0ex}}(\mathsf{\mu}$m) | 86.09 | |

${\tau}_{F}\phantom{\rule{3.33333pt}{0ex}}(\mathsf{\mu}$s) | 79.11 | |

Da | 4.82 | |

Ka | 0.44 |

**Table 2.**Characteristics of the distributions of the heterogeneities considered in the DNS database simulations. Each configuration is designated by an identifier in the form $XY$, where X is associated to the characteristic size of heterogeneous pockets (N for the homogeneous case, M for moderate size and B for relatively high size), while Y represents the distribution segregation rate (N for absence of heterogeneities, M for moderate segregation rate, and H for high segregation rate).

NN | MM | MH | BM | BH | |
---|---|---|---|---|---|

${S}_{\Phi}$ | - | 0.4 | 0.8 | 0.4 | 0.8 |

${L}_{\Phi}/{l}_{t}$ | - | 0.5 | 0.5 | 1.0 | 1.0 |

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**MDPI and ACS Style**

Er-raiy, A.; Boukharfane, R.; Parsani, M. Effects of Composition Heterogeneities on Flame Kernel Propagation: A DNS Study. *Fluids* **2020**, *5*, 152.
https://doi.org/10.3390/fluids5030152

**AMA Style**

Er-raiy A, Boukharfane R, Parsani M. Effects of Composition Heterogeneities on Flame Kernel Propagation: A DNS Study. *Fluids*. 2020; 5(3):152.
https://doi.org/10.3390/fluids5030152

**Chicago/Turabian Style**

Er-raiy, Aimad, Radouan Boukharfane, and Matteo Parsani. 2020. "Effects of Composition Heterogeneities on Flame Kernel Propagation: A DNS Study" *Fluids* 5, no. 3: 152.
https://doi.org/10.3390/fluids5030152