Self-wrinkling induced by Darrieus-Landau instability in turbulent premixed Bunsen flames from low to moderately high Reynolds numbers

Experimental data obtained via particle image velocimetry are used to investigate the self-wrinkling of premixed flame fronts induced by Darrieus-Landau (DL) instability and its interaction with turbulence from low to moderately high Reynolds numbers in a Bunsen configuration. At low Reynolds, hence in quasilaminar conditions, the DL instability is experimentally triggered by varying the mixture ratio. Conversely, in turbulent cases, the DL instability is triggered by varying the Bunsen nozzle diameter so that flames can be compared at the same equivalence ratio and jet Reynolds numbers. The differences between stable and unstable Bunsen flames at low Reynolds number are discussed in terms of vorticity generation downstream in the flame as well as total flame strain and its normal and tangential components. The straining pattern of a single DL cusp is calculated along the flame front to create a reference case for higher turbulence intensity cases and assess the influence of self-wrinkling of the flame front on the reactant flow field, i.e., the channeling effect. The results obtained are consistent with recent direct numerical simulations of single DL-cusp propagating in a quiescent environment. In addition, inspection of strain-curvature joint probability density function, curvature correlation coefficient and crossing length statistics will show that, even from the intermediate Reynolds numbers explored, unstable flames under the influence of self-wrinkling effects possess the statistical characteristics that stable flames gain only at high turbulence levels, where a unified and turbulence dominated regime is likely to be reached, although with the persistence of some residual differences.

Figure 1

Borghi-Peters diagram. Open symbols, $D=9$ mm, filled symbols, $D=18$ mm. Constant Karlovitz scaling laws follow from $u_0^{\prime }/S_L^0=K{a}^{2/3}{(\ell /{\delta }_T)}^{1/3}$.

Figure 2

Low-$Re$ Bunsen flames $D=14$ mm: Mie scattering images of the two cases and mean progress variable $\overline{c}$. Isolines represent progress variables values of $\overline{c}=0.1$, 0.5, 0.8 for both cases.

Figure 3

Vector field visualization of the low-$Re$ Bunsen flames obtained from the instantaneous velocity by subtracting half of the two-dimensional ensemble-averaged velocity flow field. Vectors are shown using half of the actual resolution and a reference vector of 5 m/s is also reported. The flame front is colored by the local flame-strain values. For the DL-unstable flame the velocity streamlines are shown. Flame strain and curvature are also displayed as functions of the local coordinate $ds$ following the flames from $A$ to $B$.

Figure 4

Detail of the streamline channeling effect induced by a singe DL-cusp as seen from the flame front in the unstable $\varphi =1.5$ case.

Figure 5

Flame curvature and strain components along the cusp-like structure marked as [1] in the previous figures. (a) Nondimensional quantities as functions of the flame coordinate $ds=0-5$ mm. (b) Total strain as a function of the flame curvature along the cusp.

Figure 6

Joint p.d.f. distribution between the components of flame strain and the flame front curvature $k$. The colormap refers to the DL-unstable case, while the black isocontour represents the DL-stable case. The marginal distributions of both $K_S^n$ and $K_S^{\tau }$ are also reported.

Figure 7

Instantaneous vorticity fields ${\omega }_z$ in the laboratory frame of reference, with two reference streamlines highlighted for each case. The bottom panels show the vorticity along the reference streamlines using the same color coding.

Figure 8

Binarized images of instantaneous flame fronts for the DL-unstable cases ($D=18$ mm) and DL-stable cases ($D=9$ mm) at increasing $Re$ with time-averaged isolines representing $\overline{c}=0.1$, 0.5, 0.9. The global consumption speed $S_T/S_L^0$ of each flame is also reported with the relative DL enhancement for the unstable cases.

Figure 9

Joint p.d.f.s between flame strain and curvature for the DL-unstable cases ($D=18$ mm) and DL-stable cases ($D=9$ mm) at increasing $Re$. Superimposed on the DL-unstable cases is $K_S$ vs $\kappa$ from the quasilaminar single-cusp case shown in Fig. 5.

Figure 10

Joint p.d.f.s between normal component of flame strain $K_S^n$ and curvature for the DL-unstable cases ($D=18$ mm) and DL-stable cases ($D=9$ mm) at increasing $Re$.

Figure 11

Joint p.d.f.s between tangential component of flame strain $K_S^{\tau }$ and curvature for the DL-unstable cases ($D=18$ mm) and DL-stable cases ($D=9$ mm) at increasing $Re$.

Figure 12

Correlation coefficient of flame-front curvature as a function of the separation $ds$ along the flame coordinate, normalized by the Bunsen diameter.

Figure 13

Turbulent flux of the averaged/filtered progress variable trough the flame brush for DL-stable ($D=9$ mm) and DL-unstable ($D=18$ mm) flames.

Figure 14

Mie-scattered light intensity as measured along portion of the mean isoline $\overline{c}=0.5$ in the 9 mm Bunsen flame at $Re=10000$. Values are purposely normalized between 0 (reactants) and 1 (products), $I_{*}=(I-I_{\min })/(I_{\max }-I_{\min })$, with $I$ being the scattered-light intensity. The gray rectangle represents the laminar unstrained flame front thermal thickness, which is on the order of 1 mm.

Figure 15

P.d.f. of flame crossing lengths of reactants for DL-unstable and DL-stable flames at increasing Reynolds numbers. Values are normalized by the corresponding mean crossing length $\overline{s}$. Also reported are the reference exponential (dashed line) and gamma-two (solid line) distributions.

Figure 16

P.d.f. of flame crossing lengths of products for DL-unstable and DL-stable flames at increasing Reynolds numbers. Values are normalized with the corresponding mean crossing length $\overline{s}$. Also reported are the reference exponential (dashed line) and gamma-two (solid line) distributions.

Figure 17

Difference between product and reactant crossing lengths, ${\overline{s}}_p-{\overline{s}}_r$, as function of $Re$ for both DL-stable and DL-unstable flames.

Figure 18

Averaged values over all the available flame images as function of the Reynolds number for both the DL-unstable and DL-stable flames at the reference progress variable ${\overline{c}}^{*}=0.5$: (a) Flame crossing per unit length ${\nu }_y$. (b) Correlation length $L_c$. (c) Evaluation of the constant $g^{\prime }={\nu }_c{L}_c/\left[\overline{c}(1-\overline{c})\right]$.

Figure 19

Comparison of FSD evaluated as flame surface to embedding volume and $\mathrm{\Sigma }={\nu }_y/{\sigma }_y$ evaluated at the reference progress variable ${\overline{c}}^{*}=0.5$.