# A Generalised Series Model for the LES of Premixed and Non-Premixed Turbulent Combustion

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## Abstract

**:**

_{4}, CO

_{2}, H

_{2}O, and O

_{2}. However, the concentrations of the intermediates CO and H

_{2}are over-predicted, due to the limitations of the reduced reaction mechanism employed. Then, a Bunsen-piloted flame is simulated. Most of the statistical properties of both the reactive species and progress variables are well reproduced. The only major discrepancy evident is in the temperature, which is probably attributed to the experimental uncertainties of temperature fields in the pilot stream. These findings demonstrate the model’s generality for both a premixed and non-premixed combustion simulation, as well as the accuracy of prediction of reactive species distribution.

## 1. Introduction

## 2. Methodology

#### 2.1. Mathematical Formulation

#### 2.2. Numerical Implementation and Error Analysis

## 3. Results and Discussion

#### 3.1. Non-Premixed Combustion Simulation

_{2}H

_{2}, H

_{2}, air, CO

_{2}, and N

_{2}with the same nominal enthalpy and equilibrium composition as methane/air, at the same equivalence ratio, at a bulk velocity of 11.4 m/s. Surrounding the pilot nozzle, air streams at a velocity of 0.9 m/s. The burner exit is positioned approximately 15 cm above the exit of the vertical wind tunnel [34].

_{2}distribution, the predicted flame structure displays the conventional characteristics of non-premixed combustion [2] as expected: a preliminary area close to the inlet nozzle where flames are thin and a subsequent zone beside which hot products fill the majority of the downstream realm.

_{4}and O

_{2}tend to be consumed faster than the experimental data indicated, consistent with the slightly over-predicted temperature trend. The predictions of CO

_{2}and H

_{2}O agree well with the measurements, though the coarse grid results are less accurate. Nevertheless, the prediction of CO and H

_{2}is significantly over-predicted. The discrepancies as well as the accelerated CH

_{4}decay rate are like those observed in the simulation [43] and can be attributed to the limitations of the simplified mechanism [42]. These findings [43,59] suggest that the C1 scheme in the mechanism [42] gives rise to an over-prediction of the reaction rates on the fuel-rich side of the non-premixed flames. This is evidenced in the radial distribution of CH

_{4}in Figure 4. Additionally, the reduced reaction mechanism suffers from the shortcomings of predicting intermediates like H

_{2}and CO as it is susceptible to diffusive transport [42]. This accounts for the inferior distributions of the same intermediate species in Figure 3. In spite of this, good radial agreement is achieved for both the series model and stochastic fields method [43], regarding the mean and RMS of CO

_{2}and H

_{2}O. The peak locations near the nozzle exit are slightly under-predicted in the reference case [43], owing to the same reason of the temperature distribution.

#### 3.2. Premixed Combustion Simulation

_{4}and O

_{2}are reasonably replicated; in spite of some minor under-predictions on the fuel-lean sides (r/H > 0.5), the calculated reactants meet the measurements well at the fuel-rich side and keep the descending trend from upstream to downstream. This indicates that the consummation rates of CH

_{4}and O

_{2}are well-reproduced along the centreline. In contrast, the reference results [22] are under-predicted on both the fuel-lean and fuel-rich sites.

_{2}and H

_{2}O by the series model show a reasonably decent consistency with experimental observations, despite some under-predictions at z/H = 6.5. Meanwhile, improvements are obvious with the increase in grid resolutions. Compared with the reference case [22], the series model performs better on the fuel-rich sides. This is because the profiles of the reactants CH

_{4}and O

_{2}are not well resolved in these areas [22]. Note that in ATF models, although the thickened flames are solved, the species transport equation is modified and the interaction between combustion and turbulence is transformed from a transport-dominant combustion regime to a chemistry-dominant one, and the impact of the heat release upon the flow field is not represented sufficiently. In contrast, the series model operates on the reaction rate term directly, without altering the formation of the species balance equations. In terms of CO, the series model also obtains a good prediction, while the profiles are slightly under-predicted in the reference case [22]. On the other hand, the computed CO products are less sensitive to grid resolutions than major species, as less difference is found with grid refinement. The measurement error of major species varies from 8% to 15% and that of intermediate species reaches 25% [26]. The major and intermediate species predicted by the series model principally cater to the accuracy in both grid resolutions.

## 4. Conclusions

_{4}, CO

_{2}, H

_{2}O, and O

_{2}are shown to be in reasonably well agreement with the experimental measurements. The discrepancies appear in the centreline and with the radial distribution of the intermediates CO and H

_{2}; this over-prediction is probably due to the limitations of the reduced reaction mechanism employed in the simulation. In contrast to the stochastic field approach using the same mechanism, the series model demonstrates comparatively good prediction in general, and is even better on some occasions, like when correctly capturing the radial distributions of mixture fractions and the peak locations of the mean temperature and chemical species.

_{4}, O

_{2}, CO

_{2}, H

_{2}O, and CO and progress variables are well reproduced by the series model. The major discrepancy lies in the temperature, which is attributed to the experimental uncertainties of the temperature fields in the pilot stream and the adiabatic boundary condition. Compared to the artificially thickened flame model that adopts a similar series approach to determine the F factor, the predictions are as good, if not better. Considering that the reference simulation employs a finer mesh and more detailed chemistry, the series model shows the potential improvement when finer meshes or/and more detailed chemistry are used.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

$c$ | Species mass concentration | $\alpha $ | Chemical species |

${C}_{sgs}$ | Sub-grid coefficient | $\mathsf{\Delta}$ | Filter size |

D | Sandia flame D inlet diameter | ${\phi}_{k}$ | Field scalar |

Da | Damköhler number | $\chi $ | Scalar dissipation rate |

hrs | Hours | $\dot{\omega}\left(c\right)$ | Chemical source term |

H | Bunsen flame F3 inlet diameter | x_{j} | The spatial vector |

r | Radial offset | Y | Species mass fraction |

Re | Reynolds number | z | The spatial z-direction vector |

T | Temperature |

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**Figure 1.**A sketch of the simulation domain and fine grid allocation. Blue area: main jet. Red area: pilot stream.

**Figure 3.**Centreline trends of the mixture fraction, temperature, axial velocity, and reactive species. Black solid line: mean value of the series model with a fine grid (i). Blue solid line: mean value of the series model with a coarse grid (ii). Square scatter: mean experimental data [34] (iii). Black dashed line: rms value of the series model with a fine grid (iv). Blue dashed line: rms value of the series model with a coarse grid (v). Triangle scatter: rms experimental data [34] (vi). Green solid line: mean value in reference [43] (vii). Green dashed line: rms value in reference [43] (viii).

**Figure 4.**Radial distributions of the mixture fraction, temperature, and reactive species. Black solid line: mean value of the series model with a fine grid (i). Blue solid line: mean value of the series model with a coarse grid (ii). Square scatter: mean experimental data [34] (iii). Black dashed line: rms value of the series model with a fine grid (iv). Blue dashed line: rms value of the series model with a coarse grid (v). Round scatter: rms experimental data [34] (vi). Green solid line: mean value in reference [43] (vii).

**Figure 5.**A sketch of the simulation domain and grid allocation. Blue area: main jet. Red area: pilot stream.

**Figure 7.**Progress variable radial profiles. Black solid line: series model with a fine grid (i). Blue solid line: series model with a coarse grid (ii). Square scatter: experimental data [26] (iii).

Research | Turbulent SGS Closures | Turbulent Reacting LES Closures | Simulation Domain | Grid Resolution | Chemistry Mechanism |
---|---|---|---|---|---|

Current study | Dynamic eddy viscosity model | Series model | (15~30D) × 2π × 70D | Fine: 71 × 48 × 210 + 12 × 12 × 210 Coarse: 52 × 36 × 139 + 9 × 9 × 139 (Polar coordinates + o-grid) | Jones–Lindstedt four-step mechanism |

[43] | Eddy viscosity model | Eulerian stochastic field method | 40D × 40D × 84D | 68 × 68 × 106 (Cartesian coordinates) | Jones–Lindstedt four-step |

[44] | SIGMA eddy viscosity model | Direct integration of chemical kinetics | 40D × 40D × 138D | 375 million tetrahedral elements (unstructured meshes) | GRI 2.0 and 3.0 |

[45] | Dynamic Smagorinsky | Multi-environment PDF model | (8~44D) × 2π × 80D | 101 × 64 × 197 (cylindrical coordinates) | Reduced GRI 3.0 |

[46,47] | Eddy–viscosity model | Presumed β-pdf and Thickened flame approach | 40D × 40D × 150D | 128 × 128 × 320 (Cartesian coordinates) | GRI 3.0 |

[48,49] | Dynamic Smagorinsky | Extended flamelet/progress variable model | 26.5D × 2π × 80D | 160 × 64 × 256 (cylindrical coordinates) | GRI 2.11 |

[36] | Smagorinsky | Eulerian stochastic field method | 20D × 20D × 50D | 81 × 81 × 160 (Cartesian coordinates) | Augmented GRI3.0 |

[50] | Dynamic Smagorinsky | Lagrangian filtered-density approach | 20D × 2π × 80D | 256 × 128 × 32 (cylindrical coordinates) | GRI-2.11 |

[51] | Dynamic Smagorinsky | Conditional Moment Closure | 20D × 20D × 80D | 1.3M nodes (CMC grids) | ARM2 chemistry |

[52,53,54] | Dynamic Smagorinksy | Hybrid Eulerian–Lagrangian MMC model | 35D × 2π × 35D | 512 × 55 × 32 (cylindrical coordinates) | GRI-3.0 |

[55] | One equation eddy viscosity | Eddy Dissipation Concept | 21D × 2π × 73D | 240 × 60 × 90 (cylindrical coordinates) | GRI3.0 and Single-step mechanism |

[56] | Smagorinsky | Lagrangian Flamelet Model | 15D × 2π × 80D | 110 × 48 × 192 (cylindrical coordinates) | GRI 2.11 |

[57] | Modified kinetic energy viscosity | Flamelet model | 15D × 15D × 80D | 101 × 101 × 91 (Cartesian coordinates) | GRI 2.11 |

[58] | Smagorinsky | Conditional Moment Closure | 8D × 8D × 80D | 96 × 96 × 320 (Cartesian coordinates) | Meyer mechanism |

Research | Turbulent SGS Closures | Turbulent Reacting LES Closures | Simulation Domain | Grid Resolution | Chemistry Mechanism |
---|---|---|---|---|---|

Current | Dynamic eddy viscosity model | Series model | 12H × 2π × 30H | Fine: 69 × 48 × 200 + 12 × 12 × 200 Coarse: 49 × 36 × 134 + 9 × 9 × 134 (Polar coordinates + o-grid) | Jones and Lindstedt’s four-step mechanism |

[22,23] | Vreman model | Artificially thickened flame | 8H × 8H × 16H | 194 × 194 × 306 (Cartesian coordinates) | GRI 3.0 |

[61,62] | Smagorinsky | G-field | 4H × 4H × 20H | 64 × 64 × 296 (Cartesian coordinates) | GRI-MECH 2.11 |

[63] | Germano model | G-field and dynamic propagation model | 6H × 6H × 30H | 117 × 64 × 323 (cylindrical coordinates) | GRI |

[60] | Smagorinsky | Eulerian stochastic fields | 5H × 5H × 15H | 56 × 36 × 112 (Cartesian coordinates) | ARM for NO |

[64,65] | Dynamic Smagorinsky | Artificially thickened flame | 4H × 2π × 20H | 94 × 64 × 300 (cylindrical coordinates) | A two-step mechanism |

[66,67,68] | Smagorinsky | Dynamic modelling and Assumed PDF | 20H × 20H × 40H | 1.5 minion cells (Cartesian coordinates) | Augmented reduction of GRI3.0 |

[69] | Dynamic Smagorinsky | Dynamic thickened flame | 40H × 40H × 120H | Unstructured meshes | A single-step mechanism |

[6] | Smagorinsky | Dynamic thickened flame model | 40H × 40H × 120H | Unstructured meshes | A two-step mechanism |

[70] | Second moment | Transported pdf | 4H × 4H × 12.5H | Lagrangian particle grids | Lindstedt reduced mechanism |

[71] | Linear stress model | Pdf method | 6.5H × 20H | 70 × 220 (2D simulation) | Drm22 |

[72] | Smagorinsky | G-equation | 6H × 6H × 45H | 345,000 cells (cylindrical coordinates) | Schmidt mechanism |

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## Share and Cite

**MDPI and ACS Style**

Zeng, W.; Wang, X.; Luo, K.H.; Vogiatzaki, K.; Navarro-Martinez, S.
A Generalised Series Model for the LES of Premixed and Non-Premixed Turbulent Combustion. *Energies* **2024**, *17*, 252.
https://doi.org/10.3390/en17010252

**AMA Style**

Zeng W, Wang X, Luo KH, Vogiatzaki K, Navarro-Martinez S.
A Generalised Series Model for the LES of Premixed and Non-Premixed Turbulent Combustion. *Energies*. 2024; 17(1):252.
https://doi.org/10.3390/en17010252

**Chicago/Turabian Style**

Zeng, Weilin, Xujiang Wang, Kai Hong Luo, Konstantina Vogiatzaki, and Salvador Navarro-Martinez.
2024. "A Generalised Series Model for the LES of Premixed and Non-Premixed Turbulent Combustion" *Energies* 17, no. 1: 252.
https://doi.org/10.3390/en17010252