# Dynamical Behavior of Small-Scale Buoyant Diffusion Flames in Externally Swirling Flows

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Computational Methodology

## 3. Results and Discussion

- How finite rate chemistry affects a flickering buoyant diffusion flame;
- How a flickering buoyant diffusion flame exhibits when the surrounding airflow is swirling;
- Why the flame flicker vanishes once the swirling intensity increases to a certain degree;
- What the vortex-dynamical interpretations for the flame variation under different swirling conditions are.

#### 3.1. Finite Rate Chemistry Effects on Flame Flicker

^{2}in the quiescent environment for the range of $Re=100~120$ and $Fr=0.05~0.56$. Figure 4 shows that the flickering frequencies of diffusion flames in the low Froude regime ($Fr<1$) agree well with previous experiments [22,26,61,62]. The scaling relations of $St=0.29{Fr}^{-0.5}$ and $St=0.56{Fr}^{-0.5}$ are available for flickering jet flames [27] and puffing pool fires [63], respectively, at a small Froude number regime. In addition, the comparisons between the reaction mechanisms show that the reaction has slight influences on the flickering frequency when the environment airflow is quiescent. The observation is consistent with the finding that the fuel types and the chemistry have an insensitive influence on the flame flicker [64,65,66]. Considering that the local extinction tends to occur with increasing intensities of the swirling flow, we used one-step finite-rate chemistry in all the simulations in the following analysis.

#### 3.2. Flickering Flames: Benchmark Cases

^{2}) in a quiescent environment (i.e., no externally swirling flow, R = 0), where the infinitely fast chemistry (IFC) assumption is used in Case I, whereas the one-step finite-rate chemistry (OFC) assumption is used in Case II. Due to the instability of flame-induced buoyancy [30,67], an axis-symmetric toroidal vortex is formed as the growth and roll-up of shearing between the flame sheet and the surrounding air, as shown in Figure 5a,b. It should be noted that the flickering flame is in varicose mode (outer buoyancy-induced shear layer is dominated) [67,68] and always keeps axial symmetry (the streamlines always stay in the plane crossing the central axis) during the up-and-down periodic motion.

#### 3.3. Faster Flickering Flames

#### 3.4. Oscillating Flames

#### 3.5. Steady Flames

#### 3.6. Lifted Flame

#### 3.7. Spiral Flame

#### 3.8. Vortex Bubble Flame

## 4. Concluding Remarks

- The flickering flames have the distinct feature that the periodic shedding of the toroidal vortex around the flame. The portraits of these flames are the closed ring shape. Additionally, the topological structure of the flames is broken when the externally swirling flow is weak, for instance, the weak swirling conditions of $R<$ 0.31 and $\alpha =45\xb0$ in this study.
- The oscillating mode exhibits that the toroidal vortex sheds off behind the flame and occurs at the intermediate $R$ region (for instance, $R=0.29~0.35$ and $\alpha =45\xb0$ in this study). The upstream portrait of these oscillating flames is the closed ring, while a big disturbance occurs in the downstream portrait.
- The steady mode hardly has the formation of a toroidal vortex around the flame, as the vortex shedding occurs far behind the flame. In the steady flames, the upstream phase portrait degenerates into a point, while the downstream portrait exhibits small oscillation. The formation of steady flames corresponds to the relevantly large $R$ region, for instance, $0.35<R<1.11$ and $\alpha =45\xb0$ in this study.
- The lifted flames detach from the bottom wall due to the relatively small $Da$ number. The phase portraits of the flames are nearly motionless. The present study shows that the large $R$ (>1.10) with the fixed $\alpha =45\xb0$ causes a very small ratio of the residence time to the chemical time at the flame base, thereby leading to the lift-off of the flame.
- The spiral flames have a distinct feature in that the symmetry of shear layers around the flame is broken, compared with the four modes of flickering, oscillating, steady, and lifted flames. In these flames, the upstream phase portrait is a small ellipse, while the downstream portrait shows a big quasi-cycle. The asymmetric flames occur at a large $\alpha $, while $R$ is the same. For instance, $R$ = 0.60 and $\alpha $ = 79° in this study.
- The vortex bubble flames show a different pattern in the occurrence of the vortex bubble for the vortex breakdown in the flame base, compared with the lifted flame. The phase portraits present a warping string within a relatively small range as the unstable bubble has time-varying barycenter and shape. These flames occur at the relatively large $R$ and $\alpha $; for instance, $R$ = 1.30 and $\alpha $ = 64° in this study.

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**(

**a**) Schematic of the present simulations including the domain, mesh, and boundaries. (

**b**) The swirling flow is adjusted by the four wind walls with inlet velocity $U=\left({U}_{\perp},{U}_{\parallel}\right)$, where $\alpha $ is the included angle between velocity components and $R=U/{U}_{0}.$ The contour of the Y−Z plane shows the vertical component ${\widehat{\omega}}_{z}$ of vorticity.

**Figure 3.**(

**a**) The flow field (velocity vector and streamline) of the X−Y planes at $\widehat{z}=$ 3, 6, and 9 for the case at $R=0.17$ and $\alpha =45\xb0$. (

**b**) The radial profiles of azimuthal velocity ${\widehat{u}}_{\theta}$. A vortex core is formed within the radial location ${\widehat{r}}_{a}$, where increases monotonously up to the maximum ${\widehat{u}}_{\theta ,max}$. Eight azimuth angles that are distributed with $45\xb0$ differences are denoted by different geometrical symbols. (

**c**) The correlation between the circulation $\widehat{\mathsf{\Gamma}}$ and the swirling intension $R$with the fixed $\alpha =45\xb0$. The circulation is defined as $\widehat{\mathsf{\Gamma}}=\int \mathit{u}d\mathit{l}/\sqrt{g{D}^{3}}$along the closed circle $\mathit{l}$ with the radius ${\widehat{r}}_{a}$ at three cross-sections of $\widehat{z}=3$, 6, and 9.

**Figure 5.**The contour of vorticity ${\widehat{\omega}}_{\theta}$ of flickering buoyant diffusion flames for the benchmark cases (${U}_{0}=$ 0.165 m/s, $Re$ = 100, $Fr$ = 0.28, and no swirling flow) in Figure 2: (

**a**) Case I (IFC) and (

**b**) Case II (OFC). The flame is represented by the orange isoline of heat release. The streamlines are plotted around the flame. (

**c**,

**d**) The time-varying evolution of flames and vortices in the benchmark cases.

**Figure 6.**(

**a**,

**b**) Six axial velocities ${S}_{{U}_{i}},i=$ 1, 2, 3, 4, 5, and 6 at $\widehat{z}=$ 3, 6, 9, 12, 15, and 18, respectively, along the central axis in the benchmark cases shown in Figure 5. (

**c**–

**f**) Their phase portraits in the cubic space with the same range of ${C}_{\rho}\sqrt{gD}$, where ${C}_{\rho}={\rho}_{\infty}/{\rho}_{f}=7.5$ is the density ratio of ambient air and flame.

**Figure 7.**Faster flicker of the buoyant diffusion flame ($Re$ = 100, $Fr$ = 0.28) at $R=0.26$ with the fixed $\alpha =45\xb0$: (

**a**) the flow around the flame, (

**b**) the time-varying evolution of the flame, (

**c**,

**d**) the phase portrait in the cubic space plotted by six velocity components ${S}_{{U}_{i}},i=$ 1, 2, 3, 4, 5, and 6 at $\widehat{z}=$ 3, 6, 9, 12, 15, and 18, respectively, along the central axis. All phase spaces have the same range of ${C}_{\rho}\sqrt{gD}$. The benchmark flame at $R=0$ is shown in the dotted rectangle.

**Figure 8.**Comparison between the correlation of $\u2206\widehat{f}=(\widehat{f}-{\widehat{f}}_{0})~{R}^{2}$ with the numerical results of the infinitely fast chemistry (IFC) and the one-step finite rate chemistry (OFC) in the present study. The swirling flows are fixed at $\alpha =45\xb0$.

**Figure 9.**Tip oscillation of the buoyant diffusion flame ($Re$ = 100, $Fr$ = 0.28) at $R=0.31$ with the fixed $\alpha =45\xb0$: (

**a**) the flow around the flame, (

**b**) the time-varying evolution of the flame, (

**c**,

**d**) the phase portrait in the cubic space plotted by six velocity components ${S}_{{U}_{i}},i=$ 1, 2, 3, 4, 5, and 6 at $\widehat{z}=$ 3, 6, 9, 12, 15, and 18, respectively, along the central axis. The upstream phase space is twice as large as the downstream is. The benchmark flame at $R=0$ is shown in the dotted rectangle.

**Figure 10.**Steady-state of the buoyant diffusion flame ($Re$ = 100, $Fr$ = 0.28) at $R=0.43$with the fixed $\alpha =45\xb0$: (

**a**) the flow around the flame, (

**b**) the time-varying evolution of the flame, (

**c**,

**d**) the phase portrait in the cubic space plotted by six velocity components ${S}_{{U}_{i}},i=$ 1, 2, 3, 4, 5, and 6 at $\widehat{z}=$ 3, 6, 9, 12, 15, and 18, respectively, along the central axis. All phase spaces have the same range of ${C}_{\rho}\sqrt{gD}$. The benchmark flame at $R=0$ is shown in the dotted rectangle.

**Figure 11.**The vertical position of vortex shedding-off ${H}_{v}$ vs. the maximum flame height ${H}_{f}$ at the fixed $\alpha =45\xb0$ but different $R$.

**Figure 12.**Lift-off of the buoyant diffusion flame ($Re$ = 100, $Fr$ = 0.28) at $R=1.11$ with the fixed $\alpha =45\xb0$: (

**a**) the flow around the flame, (

**b**) the time-varying evolution of the flame, (

**c**,

**d**) the phase portrait in the cubic space plotted by six velocity components ${S}_{{U}_{i}},i=$ 1, 2, 3, 4, 5, and 6 at $\widehat{z}=$ 3, 6, 9, 12, 15, and 18, respectively, along the central axis. All phase spaces have the same range of ${C}_{\rho}\sqrt{gD}$. The benchmark flame at $R=0$ is shown in the dotted rectangle.

**Figure 13.**The variation of circulation with time during the lift-off formation from the attached flame. The circulation is defined as $\widehat{\mathsf{\Gamma}}=\int \mathit{u}d\mathit{l}/\sqrt{g{D}^{3}}$along the closed circle $\mathit{l}$ with the radius $\widehat{r}=3$. The two lines represent the cross-section at $\widehat{z}=1$ and 3, respectively. Six instantaneous snapshots of flame and vorticity are included.

**Figure 14.**Spiral structure of the buoyant diffusion flame ($Re=120$, $Fr$ = 0.40) in the swirling flow with $R=0.60$ and $\alpha =79\xb0$. (

**a**) the flow around the flame, (

**b**) the time-varying evolution of the flame, (

**c**,

**d**) the phase portrait in the cubic space plotted by six velocity components ${S}_{{U}_{i}},i=$ 1, 2, 3, 4, 5, and 6 at $\widehat{z}=$ 3, 6, 9, 12, 15, and 18, respectively, along the central axis. All phase spaces have the same range of ${C}_{\rho}\sqrt{gD}$. In the dotted rectangle, the oscillating flame ($Re=120$, $Fr$ = 0.40) at $R=0.60$ and $\alpha =45\xb0$ is shown to facilitate comparison.

**Figure 15.**The three-dimensional view and their three-view drawings of (

**a**) the oscillating flame ($\alpha =45\xb0$) and (

**b**) the spiral flame ($\alpha =79\xb0$) corresponding to Figure 14, respectively. The flame is represented by the orange iso-surface of heat release. The vorticial structure is denoted by the grey iso-surface of vorticity.

**Figure 16.**Vortex bubble of the buoyant diffusion flame ($Re$ = 100, $Fr$ = 0.28) in the swirling flow with $R=1.30$ and $\alpha =64\xb0$. (

**a**) the flow around the flame, (

**b**) the time-varying evolution of the flame, (

**c**,

**d**) the phase portrait in the cubic space plotted by six velocity components ${S}_{{U}_{i}},i=$ 1, 2, 3, 4, 5, and 6 at $\widehat{z}=$ 3, 6, 9, 12, 15, and 18, respectively, along the central axis. All phase spaces have the same range of ${C}_{\rho}\sqrt{gD}$. In the dotted rectangle, the lifted flame at $R=1.30$ and α = 45° is shown to facilitate comparison.

**Figure 17.**Comparison of the flame with vortex bubble and the simulated blue whirl (insert figure) [73]. The streamlines are colored by the temperature and the flame is plotted by the iso-surface of the heat release rate at 1 MW/m

^{3}.

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Yang, T.; Ma, Y.; Zhang, P.
Dynamical Behavior of Small-Scale Buoyant Diffusion Flames in Externally Swirling Flows. *Symmetry* **2024**, *16*, 292.
https://doi.org/10.3390/sym16030292

**AMA Style**

Yang T, Ma Y, Zhang P.
Dynamical Behavior of Small-Scale Buoyant Diffusion Flames in Externally Swirling Flows. *Symmetry*. 2024; 16(3):292.
https://doi.org/10.3390/sym16030292

**Chicago/Turabian Style**

Yang, Tao, Yuan Ma, and Peng Zhang.
2024. "Dynamical Behavior of Small-Scale Buoyant Diffusion Flames in Externally Swirling Flows" *Symmetry* 16, no. 3: 292.
https://doi.org/10.3390/sym16030292