# Convective Velocity Perturbations and Excess Gain in Flame Response as a Result of Flame-Flow Feedback

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Flow Dynamics

#### 2.1.1. Governing Equations

#### 2.1.2. Conformal Mapping-Based Modeling Approach

#### 2.1.3. Flow-Field Singularities

#### 2.1.4. Kutta Condition

#### 2.2. Flame Dynamics

#### 2.2.1. Linearized G-Equation for the Flame Front

#### 2.2.2. Jump Conditions across a Flame Sheet

#### 2.2.3. Modeling of Flame Generated Vorticity

#### 2.3. Test Case Setup

## 3. Results—Analysis of Flow/Flame Interactions

#### 3.1. Unidirectional Coupling

#### 3.1.1. Harmonic Forcing

#### 3.1.2. Impulse Forcing

#### 3.2. Bidirectional Coupling

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CFD | Computational Fluid Dynamics |

CVP | Convective Velocity Perturbation |

BD | BiDirectional coupling |

FTF | Flame Transfer Function |

FR | Frequency Response |

IR | Impulse Response |

UD | UniDirectional coupling |

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**Figure 1.**A laminar slit flame is stabilized above a planar jet of unburned mixture, as illustrated in 3D in the lower left corner of the picture. The unburned (“u”) fuel/air premixture flows with bulk flow velocity ${\overline{u}}_{1,\mathrm{blk}}$ and is subjected to upstream bulk velocity perturbations ${u}_{1,\mathrm{blk}}^{\prime}$. The flame propagates normal to itself with speed ${s}_{L}$, which may depend on flame front curvature $\kappa $, and acts as a volume source of strength m. Two coordinate systems are used, namely laboratory coordinates $({x}_{1},{x}_{2})$ and flame-aligned coordinates $({x}_{\Vert},{x}_{\perp})$ where ${x}_{\perp}$ points in the local flame normal direction towards the burned gas (“b”).

**Figure 2.**Illustration of the Schwarz–Christoffel mapping (not to scale). Mean flame position ( ) and straight lines radiating from the origin of the image domain ( ), each in the physical domain (

**left**) and the image domain (

**right**).

**Figure 3.**Illustration of how impermeability boundary conditions are met by use of a mirror vortex (not to scale): the wall normal velocity component of the original vortex of strength $\Delta {\Gamma}^{\prime}$ (green) is canceled by its mirror counterpart (pink) such that the resulting velocity (red) is parallel to the walls in the physical (

**left**) as well as in the image domain (

**right**). The shear layer and the flame front are shown as blue dashed and red dotted lines, respectively.

**Figure 4.**Flow field in a combustor with ${C}_{r}=0.4$ resulting from a source at $\xi =0$ and a Kutta panel (green). Close up views of the Kutta panel ${H}_{\xi}=2.1$ and ${\beta}_{\xi}=\pi /3$ are shown in the physical (bottom left) and the image domain (bottom right).

**Figure 5.**Illustration of the line integral of Equation (40) along a path $\partial S$ enclosing a surface S.

**Figure 6.**Flame-generated vorticity at a corrugated flame sheet—indicated by green circular arrows—increases and decreases flow speed, as indicated by the red and blue areas as well as the orange stream tubes. Variations of flame speed as a result of flame stretch are indicated by downward pointing small arrows of various lengths.

**Figure 7.**CFD steady state absolute velocity for a confinement ratio of ${C}_{r}=0.4$ (upper half) and ${C}_{r}=0.66$ (bottom half). For both configurations, the location of maximum heat release ( ), the analytically predicted mean flame front ( ) and the approximate location of the shear layer ( ) are also shown.

**Figure 8.**Predictions of the UD model of a harmonically forced flame (80 Hz, $2\%\phantom{\rule{3.33333pt}{0ex}}{\overline{u}}_{\mathrm{blk}},{C}_{r}=0.4$). Shown are four snapshots taken at phases from ${0}^{\circ}$ to ${270}^{\circ}$ relative to the forcing signal. Displacement of the flame front ( ) is scaled by a factor of 7. Top: vortical flow component, with color along the shear layer indicating the strength of vorticity fluctuations. Bottom: irrotational flow component. The flame normal velocity is represented by green arrows (scaled by a factor of 4 for better visibility).

**Figure 9.**Flame normal velocity perturbation along the flame length resulting from Kutta conditions with panel lengths ${H}_{\xi}$ ( ), ${H}_{\xi}/2$ ( ), ${H}_{\xi}/4$ ( ) and ${H}_{\xi}/8$ ( ). ${H}_{\xi}$ is set to $2.1$ for ${C}_{r}=0.4$ (c.f. Figure 4) and to $1.8$ for ${C}_{r}=0.66$, respectively.

**Figure 10.**Four snapshots taken at equidistant instances in time of the normalized flame displacement plotted over ${x}_{\Vert}/{L}_{f}$ resulting from an impulsive velocity forcing at ${t}^{\ast}=0$ for confinement ratios ${C}_{r}=0.4$ (

**top**) and ${C}_{r}=0.66$ (

**bottom**). CFD results ( ); UD model ( ); BD model ( ).

**Figure 11.**Impulse responses (left plot in sub-figures (

**a**,

**b**)) and frequency responses (right plots in sub-figures (

**a**,

**b**)). CFD results ( ) vs. predictions of the unidirectional ( ) and bidirectional ( ) models, that is without and with consideration of flame-generated vorticity (${r}_{0,\omega ,\mathrm{min}}=4.25{l}_{M}$).

**Figure 12.**(

**a**): Downstream of a flame, velocity vectors are bent towards the flame normal direction, since the flow accelerates across the flame front. (

**b**): The instability of a flat flame sheet results from contraction (red) and expansion (blue) of flow tubes across a corrugated flame front that can be linked to consequences of flame generated vorticity (captured by the BD model) and irrotational contributions of flow expansion (not captured).

**Figure 13.**Predictions obtained with the BD model of a flame with confinement ratio ${C}_{r}=0.4$ forced harmonically at 120 Hz and an amplitude of $10\%\phantom{\rule{3.33333pt}{0ex}}{\overline{u}}_{\mathrm{blk}}$. Color raster: axial velocity ${u}_{1}^{\prime}$; mean flame front position ( ); perturbed flame ( ). Minimum vortex kernel radius ${r}_{0,\omega ,\mathrm{min}}=4.25\phantom{\rule{0.166667em}{0ex}}{l}_{M}$.

**Figure 14.**Results of a simulation as described in Figure 13, but for confinement ratio ${C}_{r}=0.66$.

Parameter | Value |
---|---|

Expansion ratio e | 6.68 |

Flame speed ${s}_{L}^{0}$ | 0.2686 $\mathrm{m}/\mathrm{s}$ |

Flame thickness ${\delta}_{D}$ | 83.95 $\mathsf{\mu}$m |

Markstein number ${M}_{a}$ | 4 |

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**MDPI and ACS Style**

Steinbacher, T.; Polifke, W.
Convective Velocity Perturbations and Excess Gain in Flame Response as a Result of Flame-Flow Feedback. *Fluids* **2022**, *7*, 61.
https://doi.org/10.3390/fluids7020061

**AMA Style**

Steinbacher T, Polifke W.
Convective Velocity Perturbations and Excess Gain in Flame Response as a Result of Flame-Flow Feedback. *Fluids*. 2022; 7(2):61.
https://doi.org/10.3390/fluids7020061

**Chicago/Turabian Style**

Steinbacher, Thomas, and Wolfgang Polifke.
2022. "Convective Velocity Perturbations and Excess Gain in Flame Response as a Result of Flame-Flow Feedback" *Fluids* 7, no. 2: 61.
https://doi.org/10.3390/fluids7020061