# Non-Premixed Filtered Tabulated Chemistry: Filtered Flame Modeling of Diffusion Flames

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Flamelet Equations

#### 2.2. Filtered Reactive Flow Governing Equations

#### 2.3. Non-Premixed Filtered Tabulated Chemistry Closure

#### 2.4. Coflow Flame Configuration

#### 2.5. Numerical Setup

#### 2.6. Filtered Chemical Database

## 3. Results

#### 3.1. Filtered Tabulated Chemistry Problem

#### Filtered Manifold Transformation

#### 3.2. Flame Sensor

#### 3.2.1. Justification

#### 3.2.2. Proposed Definition

#### 3.3. Filtered Tabulated Chemistry Results

## 4. Discussion

#### 4.1. Flame Structure

#### 4.2. Model Correction Terms Sensitivity

#### 4.3. Sensor Performance

## 5. Conclusions

- Far from the centerline the flame front does not only satisfy the counterflow hypothesis, but the flamelet identifier ${\mathrm{K}}_{\mathrm{Label}}$ remains unaltered after the filtering operation, what can be understood as the filtering of the same trajectory, namely of the same flamelet.
- The effect of a modified filtered profile at the reaction zone is then transported through convection and diffusion towards higher Z zones, and is the cause for the centerline profile modifications, where the model is not active.
- The profile extension due to the filtering process is mainly driven by the model correction terms, while the decrease in the peak values of non monotonically evolving variables depends on the filtered parameters that enter the transport equation.
- The sensor adequately identifies the multidimensional effect at the centerline, and its active range both in Z as in physical space changes throughout the domain, in accordance with the strain rate decrease and higher flame thickness.

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

Greek letters | |

${\alpha}_{\phi}$ | Unresolved diffusive contribution |

${\gamma}_{i}$ | Progress variable species i weight factor |

$\delta $ | Flame thickness |

$\lambda $ | Thermal conductivity |

$\phi $ | Specific value of a parameterizing variable |

$\rho $ | Mass density |

$\tau $ | Laminar viscous tensor |

$\dot{\omega}$ | Chemical production rate |

$\Delta $ | Filter size |

${\Delta}_{x}$ | Numerical grid spacing |

${\mathrm{\Omega}}_{\phi}$ | Unresolved convective contribution |

Latin letters | |

c | Progress variable |

${c}_{p}$ | Specific heat at constant pressure |

h | Enthalpy |

p | Pressure |

s | Spatial coordinate perpendicular to the flame front |

u | Velocity |

D | Molecular diffusivity |

K | Strain rate |

${\mathrm{K}}_{\mathrm{Label}}$ | Flamelet label |

$Le$ | Lewis number |

S | Flame sensor |

${Y}_{i}$ | Mass fraction of species i |

Z | Mixture fraction |

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**Figure 1.**Diffusion lifted flame temperature field (

**a**) and representation of the numerical grid (

**b**).

**Figure 2.**Centerline profiles of mixture fraction, normalized model correction terms and strain rate for $\u2206=2$ mm. Solid lines represent the filtered tabulated chemistry results, circles correspond to the reference solution. The convention [–] is employed to denote non-dimensional quantities.

**Figure 3.**Flamelet transformation for two different strain rates and three filter sizes. Blue lines K = 0.03 s

^{−1}and black lines K = 482 s

^{−1}. The three blue lines superimpose so that only one profile can be perceived.

**Figure 4.**Gradient angle profiles at two different radial sections. Colored lines correspond to $\nabla Z$, black lines are $\nabla {\mathrm{CO}}_{2}$.

**Figure 5.**Numerical grid resolution effect over the mixture fraction gradient for two CH4-Air flamelets with $\u2206=2$ mm. Blue line $\u2206/{\u2206}_{x}=4$, green line $\u2206/{\u2206}_{x}=2$, red line $\u2206/{\u2206}_{x}=1$, dashed lines are the reference filtered flamelet.

**Figure 6.**Filtered axial profiles of mixture fraction and progress variable (

**a**); zoom of region with bigger filter effect for Z (

**b**) and for c (

**c**). Solid lines represent the filtered tabulated chemistry results, circles correspond to the filtered reference solution.

**Figure 7.**Filtered radial profiles. Solid lines represent the filtered tabulated chemistry results, circles correspond to the filtered reference solution.

**Figure 8.**Strain rate and progress variable radial profiles in physical and mixture fraction space. Solid lines radial profiles, vertical dashed line ${Z}_{st}$.

**Figure 9.**Mixture fraction and progress variable centerline and radial profiles for three different filter sizes. The manifolds are generated by direct flamelet filtering, the filtered tabulated chemistry correction terms being omitted.

**Figure 10.**Effect of ct variation, assessed on the Z centerline profiles for three different filter sizes.

**Figure 11.**OH radial profiles for $\u2206=2$ mm at different downstream locations represented in mixture fraction (

**a**) and physical space (

**b**). (

**c**) corresponds to the limit values of (

**b**) along the axial direction. Solid lines OH, dashed lines flame sensor.

**Figure 12.**Variable transformation due to filtering for two steady state flamelets and their similitude with unfiltered unsteady flamelets when plotted in Z space. Solid lines steady profiles, dashed lines unsteady profiles.

**Figure 13.**Profile distinction for Z space analogous trajectories of steady filtered and unsteady unfiltered flamelets when observed in the physical space. Solid lines unfiltered profiles, dashed lines filtered profiles, and the red line is a separation included to better individualize the curves.

**Table 1.**Numerical grid parameters. ${\Delta}_{x,st}$ is the average cell size at the stoichiometric condition throughout the domain ${\Delta}_{x,st}=\langle {\Delta}_{x}|Z={Z}_{st}\pm 0.01\rangle $.

Mesh | NoCells | ${\mathbf{\Delta}}_{\mathit{x},\mathit{min}}$ [mm] | ${\mathbf{\Delta}}_{\mathit{x},\mathit{max}}$ [mm] | ${\mathbf{\Delta}}_{\mathit{x},\mathit{st}}$ [mm] |
---|---|---|---|---|

NU1 | 127,658 | 0.1 | 0.23 | 0.13 |

NU2 | 33,300 | 0.2 | 0.45 | 0.24 |

NU3 | 9225 | 0.4 | 0.82 | 0.45 |

Z | c | u [m/s] | p [Pa] | |
---|---|---|---|---|

Jet | 1 | 0 | parabolic profile, ${u}_{max}$0.23 | z.G. |

Coflow | 0 | 0 | 0.23 | z.G. |

Side wall | 0 | 0 | no-slip | z.G. |

Outflow | z.G. | z.G. | z.G. | 101,325 |

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**MDPI and ACS Style**

Obando Vega, P.J.; Coussement, A.; Sadiki, A.; Parente, A. Non-Premixed Filtered Tabulated Chemistry: Filtered Flame Modeling of Diffusion Flames. *Fuels* **2021**, *2*, 87-107.
https://doi.org/10.3390/fuels2020006

**AMA Style**

Obando Vega PJ, Coussement A, Sadiki A, Parente A. Non-Premixed Filtered Tabulated Chemistry: Filtered Flame Modeling of Diffusion Flames. *Fuels*. 2021; 2(2):87-107.
https://doi.org/10.3390/fuels2020006

**Chicago/Turabian Style**

Obando Vega, Pedro Javier, Axel Coussement, Amsini Sadiki, and Alessandro Parente. 2021. "Non-Premixed Filtered Tabulated Chemistry: Filtered Flame Modeling of Diffusion Flames" *Fuels* 2, no. 2: 87-107.
https://doi.org/10.3390/fuels2020006