Vortex-dynamical interpretation of anti-phase and in-phase flickering of dual buoyant diffusion flames

Anti-phase and in-phase flickering modes of dual buoyant diffusion flames were numerically investigated and theoretically analyzed in this study. Inspired by the flickering mechanism of a single buoyant diffusion flame, for which the deformation, stretching, or even pinch-off of the flame surface result from the formation and evolution of the toroidal vortices, we attempted to understand the anti-phase and in-phase flickering of dual buoyant diffusion flames from the perspective of vortex dynamics. The interaction between the inner-side shear layers of the two flames was identified to be responsible for the different flickering modes. Specifically, the transition between anti-phase and in-phase flickering modes can be predicted by a unified regime nomogram of the normalized flickering frequency versus a characteristic Reynolds number, which accounts for the viscous effect on vorticity diffusion between the two inner-side shear layers. Physically, the transition of the vortical structures from symmetric (in-phase) to staggered (anti-phase) in a dual-flame system can be interpreted as being similar to the mechanism causing flow transition in the wake of a bluff body and forming the Karman vortex street.

Figure 1

(a) Illustration of the periodic flickering process of a single pool flame. (b) Schematic of the computational domain and definition of main parameters in the dual-flame system.

Figure 2

(a) Visualization of time-varying single pool flame with $d=20\mathrm{mm}$ based on the flame sheet colored by temperature contour and vorticity on the $X-Z$ plane, $Y=0$. (b) The burning rate in time domain and frequency domain (case 3).

Figure 3

Validation of single pool flames with the scaling law [40] and experiments obtained by Schönbucher et al. [42] ($◊$), Maynard [43] ($△$), Baum and McCaffrey [44] ($\square$), and Fang et al. [45] ($▽$, $○$).

Figure 4

Snapshots of flickering flames in the anti-phase mode. (a) The brightness from the experiment [25] based on two sets of three candles, (b) the heat release, and (c) the vorticity from the present simulation based on two pool flames with $d=20\mathrm{mm}$ and $l=30\mathrm{mm}$.

Figure 5

Snapshots of flickering flames in the in-phase mode. (a) The brightness from the experiment [27] based on two candles, (b) the heat release, and (c) the vorticity from present simulation based on two pool flames with $d=20\mathrm{mm}$ and $l=10\mathrm{mm}$.

Figure 6

Comparison between instantaneous dimensionless vorticity, ${\omega }_y/{\left(g/d\right)}^{1/2}$, and dimensionless Laplacian of vorticity, $|{\nabla }^2{\omega }_y|/\left(g^{1/2}{d}^{-5/2}\right)$, in (a) in-phase and (b) anti-phase flickering modes.

Figure 7

(a) The distribution of vorticity in the $X-Z$ plane for in-phase (left) and anti-phase (right) flickering modes. The flame sheets are represented by the orange contours. The streamlines are denoted by the gray lines, and the vortex lines are colored by the helicity density, $h=\mathit{u}·\mathbf{\omega }$. (b) The area integral of helicity density, $dH/dz=\int \int hdxdy\left(x\ge 0,y\ge 0\right)$, along the $z$ direction for in-phase (black solid lines) and anti-phase (red dash-dotted lines) flickering modes. The two curves of each mode correspond to the upper and lower limits of $dH/dz$ during an entire flickering period.

Figure 8

Comparison of (a) normalized $f/f_{\mathrm{s}}$ at different $l/d$ and (b) normalized $f/f_0$ at different $\alpha G{r}^{1/2}$. The red solid and blue empty symbols correspond to in-phase and anti-phase flickering modes, respectively. The line segments mark the transition between the two distinct modes, for instance, the flame and flow patterns of which correspond to case A and case B, respectively. The circle symbols are from the experiment of Kitahata et al. [25], the diamond ones from the experiment of Nakamura et al. [26], and the square and triangle symbols from present simulation.

Figure 9

Five instantaneous (a) vorticity and (b) $Q$-criterion contours of anti-phase flickering flames ($d=2$ cm) with a 30 mm gap. The black cross denotes the position of a vortex core identified by the local maximum $Q$ criterion [60]. $h^{+}$ is the vertical height between two adjacent vortex cores.

Figure 10

The spacing ratio $l/h^{+}$ during a period for four anti-phase flickering modes, which are marked by black $\square \left(l=30\mathrm{mm}\right)$, blue $◊\left(l=20\mathrm{mm}\right)$, green $△\left(l=16\mathrm{mm}\right)$, and red $○\left(l=14\mathrm{mm}\right)$, respectively. The solid line is the theoretical value of 0.28 given by von Karman [50] and the dashed line represents the experimental value around 0.2 [61].

Figure 11

The contours of heat release for different simulation cases, showing the mode transition at different gap distances by changing the viscosity alone. The white lines are isocontours of zero vorticity, along which the ambient flow temperature is estimated.

Figure 12

The time-averaged temperature vs height along the white lines in Fig. 11 for different in-phase and anti-phase flame modes at different gap distances. The dashed lines correspond to the flame pinch-off locations, $z_{\mathrm{po}}$.

Figure 13

The vortex sheet configurations for (a) a single flame and (b) two adjacent flames of opposite vorticity.

Figure 14

The contours of (a) vorticity, (b) diffusion of vorticity, and (c) temperature for the in-phase flickering flames of Fig. 6.