# A Direct Numerical Simulation Assessment of Turbulent Burning Velocity Parametrizations for Non-Unity Lewis Numbers

^{1}

^{2}

^{*}

## Abstract

**:**

_{2}-norm of the relative error with respect to experimental data from literature for different Lewis numbers, higher turbulence intensity and thermodynamic pressure levels.

## 1. Introduction

_{2}can also be reduced by using a premixed combustion of fuels such as hydrogen, ammonia or syngas. Net zero targets by governments can be achieved by the application of low/zero-carbon fuels such as biofuel, hydrogen and ammonia, as proposed by contemporary scenario plans [1].

_{2}in the fuel induces a significant amount of differential diffusion of heat and species due to the non-unity Lewis number. Thus, the non-unity Lewis number effects cannot be ignored in the premixed combustion of High-Hydrogen-Content (HHC) fuels.

- (a)
- To assess the performances of the existing parameterizations of turbulent burning velocity ${S}_{T}$ for turbulent premixed flames with characteristic Lewis numbers significantly different from unity.
- (b)
- To illustrate the impact of the projected flame brush surface area ${A}_{L}$ evaluation on turbulent burning velocity ${S}_{T}$ for turbulent Bunsen burner flames with different characteristic Lewis numbers.

## 2. Mathematical Background

## 3. Numerical Implementation

## 4. Results and Discussion

^{th}case and ‘DNS’ and ‘Model’ superscripts are used for DNS and model expression values, respectively and $N$ is the total number of different cases considered here) of the model expressions of ${S}_{T}/{S}_{L}$ listed in Table 1 are shown in Figure 7, which shows that SK, SP, SG, SZ and SB models exhibit comparable ${E}_{DNS}$, but ${E}_{DNS}$ values are high for definitions ${A}_{L}={{\displaystyle \int}}_{V}\left|\nabla \overline{c}\right|dV$, ${A}_{L}={{\displaystyle \int}}_{V}\left|\nabla \tilde{c}\right|dV$, ${A}_{L}={A}_{\overline{c}=0.5}$ and ${A}_{L}={A}_{\tilde{c}=0.5}$. The corresponding ${E}_{DNS}$ values of the model expressions for ${S}_{T}/{S}_{L}$ listed in Table 4 are shown in Figure 8. A comparison between Figure 7 and Figure 8 reveals that the modified expressions in Table 4 significantly decrease the ${E}_{DNS}$ values when ${A}_{L}={A}_{\overline{c}=0.1}$ is used for the evaluation of ${S}_{T}/{S}_{L}$.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Schematic diagram of the computational domain; (

**b**) position of the cases on Borghi–Peters diagram.

**Figure 2.**Instantaneous isosurfaces of reaction progress variable c = 0.8 (seen from the burned gas side) coloured by non-dimensional temperature θ = (T − T

_{0})/(T

_{ad}− T

_{0}) for Le = 0.34, 0.6, 0.8, 1.0, 1.2.

**Figure 3.**Variations of ${A}_{T}/{A}_{L}$ for premixed turbulent Bunsen flame cases with $Le=0.34,0.6,0.8$, $1.0$ and $1.2$ for ${A}_{L}={\int}_{V}|\nabla \stackrel{-}{c}|dV$, ${A}_{L}={\int}_{V}|\nabla \stackrel{~}{c}|dV$, ${A}_{L}={A}_{\stackrel{-}{c}=0.1}$, ${A}_{L}={A}_{\stackrel{~}{c}=0.1}$, ${A}_{L}={A}_{\stackrel{-}{c}=0.5}$ and ${A}_{L}={A}_{\stackrel{~}{c}=0.5}$.

**Figure 4.**Variations of ${S}_{T}/{S}_{L}$ for premixed turbulent Bunsen flame cases with $Le=0.34,0.6,0.8$, $1.0$ and $1.2$ for ${A}_{L}={\int}_{V}|\nabla \stackrel{-}{c}|dV$, ${A}_{L}={\int}_{V}|\nabla \stackrel{~}{c}|dV$, ${A}_{L}={A}_{\stackrel{-}{c}=0.1}$, ${A}_{L}={A}_{\stackrel{~}{c}=0.1}$, ${A}_{L}={A}_{\stackrel{-}{c}=0.5}$ and ${A}_{L}={A}_{\stackrel{~}{c}=0.5}$ along with the predictions of SK, SP, SG, SZ and SB models (see Table 1).

**Figure 5.**Variations of ${R=(S}_{T}/{S}_{L})/({A}_{T}/{A}_{L})$ and ${{R}_{mod}=Le(S}_{T}/{S}_{L})/({A}_{T}/{A}_{L})$ for premixed turbulent Bunsen flame cases with $Le=0.34,0.6,0.8$, $1.0$ and $1.2$.

**Figure 6.**Variations of ${S}_{T}/{S}_{L}$ for premixed turbulent Bunsen flame cases with $Le=0.34,0.6,0.8$, $1.0$ and $1.2$ for ${A}_{L}={\int}_{V}|\nabla \stackrel{-}{c}|dV$, ${A}_{L}={\int}_{V}|\nabla \stackrel{~}{c}|dV$, ${A}_{L}={A}_{\stackrel{-}{c}=0.1}$, ${A}_{L}={A}_{\stackrel{~}{c}=0.1}$, ${A}_{L}={A}_{\stackrel{-}{c}=0.5}$ and ${A}_{L}={A}_{\stackrel{~}{c}=0.5}$ along with the predictions of SKL, SPL, SGL, SZL, MSB and MSKL models (see Table 4).

**Figure 7.**Variations of ${E}_{DNS}=\sqrt{\sum _{k=1}^{N}{\left|\left({x}_{k}^{DNS}-{x}_{k}^{Model}\right)/{x}_{k}^{DNS}\right|}^{2}}$ for all premixed turbulent Bunsen flame cases for ${A}_{L}={\int}_{V}|\nabla \stackrel{-}{c}|dV$, ${A}_{L}={\int}_{V}|\nabla \stackrel{~}{c}|dV$, ${A}_{L}={A}_{\stackrel{-}{c}=0.1}$, ${A}_{L}={A}_{\stackrel{~}{c}=0.1}$, ${A}_{L}={A}_{\stackrel{-}{c}=0.5}$ and ${A}_{L}={A}_{\stackrel{~}{c}=0.5}$ for the predictions of SK, SP, SG, SZ and SB models (see Table 1).

**Figure 8.**Variations of ${E}_{DNS}=\sqrt{\sum _{k=1}^{N}{\left|\left({x}_{k}^{DNS}-{x}_{k}^{Model}\right)/{x}_{k}^{DNS}\right|}^{2}}$ for all premixed turbulent Bunsen flame cases for ${A}_{L}={\int}_{V}|\nabla \stackrel{-}{c}|dV$, ${A}_{L}={\int}_{V}|\nabla \stackrel{~}{c}|dV$, ${A}_{L}={A}_{\stackrel{-}{c}=0.1}$, ${A}_{L}={A}_{\stackrel{~}{c}=0.1}$, ${A}_{L}={A}_{\stackrel{-}{c}=0.5}$ and ${A}_{L}={A}_{\stackrel{~}{c}=0.5}$ for the predictions of SKL, SPL, SGL, SZL, MSB and MSKL models (see Table 4).

**Figure 10.**Variations of ${E}_{Expt}=\sqrt{\sum _{k=1}^{N}{\left|\left({x}_{k}^{Expt}-{x}_{k}^{Model}\right)/{x}_{k}^{Expt}\right|}^{2}}$ for experimental conditions C1–C9 for the experimental database by Lipatnikov et al. [10] (see Table 5) for the predictions of SKL, SPL, SGL, SZL, MSB and MSKL models (see Table 4).

Model | Model Expression |
---|---|

SK modelKolla et al. [22] | $\frac{{S}_{T}}{{S}_{L}}=\sqrt{\frac{18{C}_{\mu}}{\left(2{c}_{m}-1\right){\beta}^{\prime}}\left\{\left(2{K}_{c}^{*}-\tau {C}_{4}\right)\left\{\frac{{u}^{\prime}l}{{S}_{L}{\delta}_{th}}\right\}+\frac{2{C}_{3}}{3}\frac{{{u}^{\prime}}^{2}}{{S}_{L}^{2}}\right\}}$ |

SP modelPeters [19] | $\frac{{S}_{T}}{{S}_{L}}=1-0.195\frac{l}{{\delta}_{z}}+\sqrt{{\left(0.195\frac{l}{{\delta}_{z}}\right)}^{2}+0.78\left\{\frac{{u}^{\prime}l}{{S}_{L}{\delta}_{z}}\right\}}$ |

SG modelGülder [20] | $\frac{{S}_{T}}{{S}_{L}}=1+0.62{\left(\frac{{u}^{\prime}}{{S}_{L}}\right)}^{0.75}{\left(\frac{l}{{\delta}_{z}}\right)}^{0.25}$ |

SZ modelZimont [21] | $\frac{{S}_{T}}{{S}_{L}}=1+0.5{\left(\frac{{u}^{\prime}}{{S}_{L}}\right)}^{0.75}{\left(\frac{l}{{\delta}_{z}}\right)}^{0.25}$ |

SB modelBradley [44] | $\frac{{S}_{T}}{{S}_{L}}=1.53{\left(\frac{{u}^{\prime}}{{S}_{L}}\right)}^{0.55}{\left(\frac{l}{{\delta}_{z}}\right)}^{0.15}L{e}^{-0.3}$ |

Case | Le | Re | Grid Size | ${\mathit{U}}_{\mathit{B}}/{\mathit{S}}_{\mathit{L}}$ | ${\mathit{u}}_{\mathit{i}\mathit{n}\mathit{l}\mathit{e}\mathit{t}}^{\text{'}}/{\mathit{S}}_{\mathit{L}}$ | $\mathit{l}/{\mathit{d}}_{\mathit{n}}$ | $\mathit{l}/{\mathit{\delta}}_{\mathit{t}\mathit{h}}$ | Ka | Da |
---|---|---|---|---|---|---|---|---|---|

A | 0.34 | 1197 | 250 × 250 × 250 | 18.0 | 1.0 | 1/5 | 5.20 | 0.45 | 5.00 |

B | 0.6 | 399 | 250 × 250 × 250 | 6.0 | 1.0 | 1/5 | 5.20 | 0.45 | 5.00 |

C | 0.8 | 399 | 250 × 250 × 250 | 6.0 | 1.0 | 1/5 | 5.20 | 0.45 | 5.00 |

D | 1.0 | 399 | 250 × 250 × 250 | 6.0 | 1.0 | 1/5 | 5.20 | 0.45 | 5.00 |

E | 1.2 | 399 | 250 × 250 × 250 | 6.0 | 1.0 | 1/5 | 5.20 | 0.45 | 5.00 |

**Table 3.**Thermochemical parameters used in the model by Kolla et al. [22].

Le | ${\mathit{K}}_{\mathit{c}}^{*}/\mathit{\tau}$ | c_{m} |
---|---|---|

0.34 | 0.52 | 0.92 |

0.60 | 0.67 | 0.87 |

0.80 | 0.71 | 0.867 |

1.00 | 0.78 | 0.825 |

1.20 | 0.79 | 0.816 |

**Table 4.**Summary of modified normalized turbulent burning velocity ${S}_{T}/{S}_{L}$ model expressions.

Model | Model Expression |
---|---|

SKL model | $\frac{{S}_{T}}{{S}_{L}}=\frac{1}{Le}\sqrt{\frac{18{C}_{\mu}}{\left(2{c}_{m}-1\right){\beta}^{\prime}}\left\{\left(2{K}_{c}^{*}-\tau {C}_{4}\right)\left\{\frac{{u}^{\prime}l}{{S}_{L}{\delta}_{th}}\right\}+\frac{2{C}_{3}}{3}\frac{{{u}^{\prime}}^{2}}{{S}_{L}^{2}}\right\}}$ |

SPL model | $\frac{{S}_{T}}{{S}_{L}}=1-0.195L{e}^{-1}\frac{l}{{\delta}_{z}}+L{e}^{-1}\sqrt{{\left(0.195\frac{l}{{\delta}_{z}}\right)}^{2}+0.78\left\{\frac{{u}^{\prime}l}{{S}_{L}{\delta}_{z}}\right\}}+\left(\frac{1-Le}{Le}\right)\frac{{u}^{\prime}/{S}_{L}}{{u}^{\prime}/{S}_{L}+1}$ |

SGL model | $\frac{{S}_{T}}{{S}_{L}}=1+\left[0.62L{e}^{-1}{\left(\frac{{u}^{\prime}}{{S}_{L}}\right)}^{0.75}{\left(\frac{l}{{\delta}_{z}}\right)}^{0.25}+\left(\frac{1-Le}{Le}\right)\frac{{u}^{\prime}/{S}_{L}}{{u}^{\prime}/{S}_{L}+1}\right]$ |

SZL model | $\frac{{S}_{T}}{{S}_{L}}=1+\left[0.5L{e}^{-1}{\left(\frac{{u}^{\prime}}{{S}_{L}}\right)}^{0.75}{\left(\frac{l}{{\delta}_{z}}\right)}^{0.25}+\left(\frac{1-Le}{Le}\right)\frac{{u}^{\prime}/{S}_{L}}{{u}^{\prime}/{S}_{L}+1}\right]$ |

MSKL model | $\frac{{S}_{T}}{{S}_{L}}=\frac{1}{Le}\sqrt{\frac{18{C}_{\mu}}{\left(2{c}_{m}-1\right){\beta}^{\prime}}\left\{\left(2{K}_{c}^{*}-\tau {C}_{4}\right)\left\{\frac{{u}^{\prime}l}{{S}_{L}{\delta}_{th}}\right\}+\frac{2{C}_{3}}{3}\frac{{{u}^{\prime}}^{2}}{{S}_{L}^{2}}\right\}+\frac{L{e}^{2}}{[{u}^{\text{'}}l/{S}_{L}{\delta}_{th}+1]}}$ |

MSB model | $\frac{{S}_{T}}{{S}_{L}}=1.53{\left(\frac{{u}^{\prime}}{{S}_{L}}\right)}^{0.55}{\left(\frac{l}{{\delta}_{z}}\right)}^{0.15}L{e}^{-0.3}+\frac{1}{[{u}^{\text{'}}l/{S}_{L}{\delta}_{th}+1]}$ |

**Table 5.**Summary of the experimental conditions from Lipatnikov et al. [10].

Conditions | Mixture | u′/S_{L} | l/δ_{th} | τ | p | Le |
---|---|---|---|---|---|---|

C1 | H_{2}-air, $\varphi =0.45$ | 2.7 | 63 | 3.7 | 1.0 bar | 0.35 |

C2 | H_{2}/O_{2}/He, $\varphi =0.45$ | 2.6 | 21 | 3.0 | 1.0 bar | 0.91 |

C3 | CH_{4}-air, $\varphi =1.0$ | 2.7 | 69 | 6.5 | 1.0 bar | 1.00 |

C4 | H_{2}-air, $\varphi =0.45$ | 4.6 | 152 | 3.7 | 3.0 bar | 0.35 |

C5 | H_{2}/O_{2}/He, $\varphi =0.45$ | 6.3 | 33 | 3.0 | 3.0 bar | 0.91 |

C6 | CH_{4}-air, $\varphi =1.0$ | 4.2 | 145 | 6.5 | 3.0 bar | 1.00 |

C7 | H_{2}-air, $\varphi =0.45$ | 7.0 | 179 | 3.7 | 5.0 bar | 0.35 |

C8 | H_{2}/O_{2}/He, $\varphi =0.45$ | 9.0 | 19 | 3.0 | 5.0 bar | 0.91 |

C9 | CH_{4}-air, $\varphi =1.0$ | 5.4 | 203 | 6.6 | 5.0 bar | 1.00 |

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Mohan, V.; Herbert, M.; Klein, M.; Chakraborty, N.
A Direct Numerical Simulation Assessment of Turbulent Burning Velocity Parametrizations for Non-Unity Lewis Numbers. *Energies* **2023**, *16*, 2590.
https://doi.org/10.3390/en16062590

**AMA Style**

Mohan V, Herbert M, Klein M, Chakraborty N.
A Direct Numerical Simulation Assessment of Turbulent Burning Velocity Parametrizations for Non-Unity Lewis Numbers. *Energies*. 2023; 16(6):2590.
https://doi.org/10.3390/en16062590

**Chicago/Turabian Style**

Mohan, Vishnu, Marco Herbert, Markus Klein, and Nilanjan Chakraborty.
2023. "A Direct Numerical Simulation Assessment of Turbulent Burning Velocity Parametrizations for Non-Unity Lewis Numbers" *Energies* 16, no. 6: 2590.
https://doi.org/10.3390/en16062590