Premixed flame stability under shear-enhanced diffusion: Effect of the flow direction

In the presence of shear-enhanced diffusion (Taylor dispersion), flame propagation is effectively anisotropic. This study focuses on the influence of the direction of a shear flow relative to the direction of propagation on the diffusional-thermal instabilities of premixed flames. The problem is addressed analytically using large activation energy asymptotics, complemented by numerical simulations, in the framework of a constant density two-dimensional model. The model, obtained by depth averaging of the governing equations in a Hele-Shaw configuration, accounts for shear-enhanced diffusion. A linear stability analysis is carried out analytically, leading to a dispersion relation involving three parameters: the Lewis number Le; the Taylor-dispersion coefficient $p$, which is proportional to the Péclet number; and the angle $\varphi$ between the direction of propagation of the unperturbed planar flame and the flow direction. Based on the dispersion relation, stability diagrams are determined in terms of the parameters, along with bifurcations curves identifying the nature of the instabilities observed. It is shown that cellular instabilities expected when $\text{Le}<1$ can now occur as a result of Taylor dispersion in $\text{Le}>1$ mixtures, provided the angle $\varphi$ exceeds a critical value approximately equal to ${75}^{\circ }$. In general, it is found that an increase in $\varphi$ from $0^{\circ }$ to ${90}^{\circ }$ has a stabilizing effect in subunity Lewis number mixtures $\text{Le}<1$ and a destabilizing effect when $\text{Le}>1$. Particular attention is devoted to the cellular long-wave instability encountered, which is found to be described by a modified Kuramoto-Sivashinsky equation. The equation involves the three aforementioned parameters and includes a dispersion term (a third-order spatial derivative) as well a drift term (first-order derivative) whenever $\varphi \ne 0^{\circ }$ and $\varphi \ne {90}^{\circ }$, which is whenever the direction of the shear flow is neither parallel nor perpendicular to the direction of flame propagation.

Figure 1

Model configuration to investigate the diffusional-thermal instability of a planar flame propagating in a channel in a direction making an angle $\varphi$ with the direction of a shear flow. A unit vector in the direction of the shear flow is ${\mathbf{i}}_{\mathbf{s}}=c\mathbf{i}-s\mathbf{j}$, where $c=cos\varphi , s=sin\varphi$, and $\mathbf{i}$ and $\mathbf{j}$ are unit vectors in the $x$ and $y$ directions, respectively. In this study, $\varphi$ is allowed to take arbitrary values in $[0^{\circ },{90}^{\circ }]$, with $\varphi =0^{\circ }$ and ${90}^{\circ }$ corresponding to flame propagation in directions parallel and perpendicular to the flow direction, respectively

Figure 2

Bifurcation curves and stability and instability regions in the $l\text{−}p$ [or more precisely $l$ versus $p^2/(1+p^2)]$ plane for selected values of the angle $\varphi$. Colored regions correspond to unstable flames and white regions to stable ones. The color scale measures the magnitude of the scaled wave number $\tilde{k}$ corresponding to the most unstable mode. The case $\varphi =0^{\circ }$ corresponds to longitudinal flame propagation (i.e., propagation parallel to the shear flow) and the case $\varphi ={90}^{\circ }$ to transverse propagation. The labels I, II, and III on the curves refer to the type of bifurcation following the terminology of [17], as explained in the text and in Fig. 5.

Figure 3

Bifurcation curves and stability and instability regions in the $l\text{−}k$ plane for selected values of the angle $\varphi$ and $p$. Here the shaded regions are stable.

Figure 4

Bifurcation curves and stability and instability regions in the $\varphi \text{−}p$ [or more precisely $\varphi$ versus $p^2/(1+p^2)]$ plane for selected values of the reduced Lewis number $l$. Colored regions correspond to unstable flames and white regions to stable ones. The color scale measures the magnitude of the scaled wave number $\tilde{k}$ corresponding to the most unstable mode. The labels I, II, and III on the curves refer to the type of bifurcation following the terminology of [17], as explained in the text and in Fig. 5.

Figure 5

Schematic illustration of the three types of bifurcation identified in Figs. 2 and 4, following the terminology of [17]. Plotted is the growth rate $\mathrm{Re}\left(\sigma \right)$ as a function of the wave number $k$ near the instability onset occurring at, say, $k=k_c$. The curves move up as we cross from stable to unstable regions in the parameter space. Type I is a finite-wavelength instability corresponding to $k_c\ne 0$, while types II and III are long-wave instabilities with $k_c=0$. Note that in a type-II bifurcation $\mathrm{Re}\left(\sigma \right)(k=0)$ remains equal to zero, while in a type-III bifurcation $\mathrm{Re}\left(\sigma \right)(k=0)$ changes sign from negative to positive as we cross from stable to unstable regions in the parameter space. Mathematically, a type-II bifurcation requires the three conditions $\mathrm{Re}\left(\sigma \right)=0, d\mathrm{Re}\left(\sigma \right)/dk=0$, and $d^2\mathrm{Re}\left(\sigma \right)/d{k}^2=0$ to be met at $k=0$, as shown in the figure. Only the first two of these conditions are required for type-I and type-III bifurcations to be satisfied at $k=k_c\ne 0$ and $k=0$, respectively, where $d^2\mathrm{Re}\left(\sigma \right)/d{k}^2<0$.

Figure 6

Bifurcation curves and stability and instability regions in the $l\text{−}p$ plane [based on the KS equation (20)] for selected values of the angle $\varphi$. The shaded regions are unstable. The blue dashed lines represent the asymptotes $p=p_1$ and $p=p_2$ specified by (25) which appear when $\varphi >arccos\left(\frac{1}{3}\right)\approx 70.{53}^{\circ }$. The red dotted lines delimit the region where ${\alpha }_4<0$, and therefore the KS equation (20) is not applicable; these appear when $\varphi >85.{35}^{\circ }$.

Figure 7

The solid $\mathrm{C}$-shaped curve represents the asymptotic values $p_1$ and $p_2$ given by (25) versus the angle $\varphi$. The domain colored in yellow represents the region where ${\alpha }_4<0$. The long-wave instability is predicted by the KS equation (20) to occur in $l>0$ mixtures for values of $\varphi$ and $p$ lying to the right of the solid $\mathrm{C}$-shaped curve excluding the yellow region. In fact, the actual region of this instability in $l>0$ mixtures is to the right of the dashed $\mathrm{C}$-shaped curve excluding the yellow region. The dashed $\mathrm{C}$-shaped curve is computed by accounting for all roots of the dispersion relation, rather than the root leading to Eq. (20).

Figure 8

Reaction rate fields $\omega$ at selected times $t$ for $\mathrm{Le}=0.7$, with $p=0$ (left) and with $p=3$ and $\varphi ={60}^{\circ }$ (right). In all computations, the values $\beta =10$ and ${\alpha }_h=0.85$ are adopted, and the scales for the horizontal $x$ axis and vertical $y$ axis are given in the bottom left corner of each panel.

Figure 9

Reaction rate fields $\omega$ at selected times $t$ for $\mathrm{Le}=1.6$ and $p=3$ with $\varphi ={90}^{\circ }$ (left) and $\varphi ={80}^{\circ }$ (right).

Figure 10

Effective burning speed $U_T$ [computed according to (27)] versus time $t$ for selected values of Le, $\varphi$, and $p$.