Front propagation in a regular vortex lattice: Dependence on the vortex structure

We investigate the dependence on the vortex structure of the propagation of fronts in stirred flows. For this, we consider a regular set of vortices whose structure is changed by varying both their boundary conditions and their aspect ratios. These configurations are investigated experimentally in autocatalytic solutions stirred by electroconvective flows and numerically from kinematic simulations based on the determination of the dominant Fourier mode of the vortex stream function in each of them. For free lateral boundary conditions, i.e., in an extended vortex lattice, it is found that both the flow structure and the front propagation negligibly depend on vortex aspect ratios. For rigid lateral boundary conditions, i.e., in a vortex chain, vortices involve a slight dependence on their aspect ratios which surprisingly yields a noticeable decrease of the enhancement of front velocity by flow advection. These different behaviors reveal a sensitivity of the mean front velocity on the flow subscales. It emphasizes the intrinsic multiscale nature of front propagation in stirred flows and the need to take into account not only the intensity of vortex flows but also their inner structure to determine front propagation at a large scale. Differences between experiments and simulations suggest the occurrence of secondary flows in vortex chains at large velocity and large aspect ratios.

Figure 1

Sketch of the setup showing the chemical solution stirred by Laplace forces induced by an electric current flowing over alternate magnets (a) and the resulting array (b) or chain (c) of counter-rotating vortices. The channel depth, vortex size, and channel width are denoted $d$, $L$, and $l$, respectively. The magnetic field and the electric current are labeled $\mathbf{B}$ and $I$.

Figure 2

Front propagation with free lateral boundary conditions, except for the side vortices. Times regularly increases from left to right while fronts propagate upwards. The channel depth is $d=2\mathrm{mm}$, the vortex width $L$ is 20 mm, and the channel width $l$ equals $5L$: $l/L=5$, $U/V_0=33.0$. Time interval between pictures is 49 s.

Figure 3

Front propagation with rigid lateral boundary conditions. Times regularly increases from left to right while fronts propagate upwards. The channel depth is $d=2$ mm, the vortex width $L$ is 20 mm, and the channel width $l$ equals the vortex width $L$: $l=L=20$ mm, $U/V_0=17.7$. Time interval between pictures is 58 s.

Figure 4

Flow field measured by PIV in a channel with rigid lateral boundary conditions. The channel depth is $d=4$ mm, the vortex width $L$ is 20 mm, and the channel width $l$ equals the vortex width $L$: $l=L=20$ mm. (a) Velocity field. (b) Superimposed intensity of the velocity field intensity obtained by extrapolation techniques. The vanishing of the flow field at the lateral boundaries makes a difference of vortex structure with free boundary conditions (Fig. 5).

Figure 5

Flow field measured by PIV in a channel with free lateral boundary conditions. The channel depth is $d=4$ mm, the vortex width $L$ is 20 mm, and the channel width $l$ amounts to $6L$. The slight distortion at the left bottom originates from an uncompensated defect in the camera frame. (a) Velocity field. (b) Superimposed intensity of the velocity field intensity obtained by extrapolation techniques.

Figure 6

Front propagation for free b.c. The vortex width is $L=20$ mm. Columns correspond to channel depths $d=2,4,6$ mm. Lines correspond to reduced vortex intensities $U/V_0$ weak ($\approx 10$), moderate ($\approx 20$), and large ($\approx 30$).

Figure 7

Reduced front velocity $V_f/V_0$ as a function of the reduced vortex intensity $U/V_0$. (a) Free b.c.: a single linear relationship is displayed whatever the channel depth $d$. (b) Rigid b.c.: concave relationships are displayed with an increasing concavity with the channel depth $d$.

Figure 8

Front propagation for rigid b.c. The vortex width is $L=20$ mm. Columns correspond to channel depths $d=2,4,6$ mm. Lines correspond to reduced vortex intensities $U/V_0$ weak ($\approx 10$), moderate ($\approx 20$), and large ($\approx 30$).

Figure 9

Profile of the fundamental mode ${\psi }_{0,0}^f$ on the channel depth direction $z$ (blue full line) and comparison with a sinusoidal shape (a) (red dashed line) and a parabolic shape (b) (red dashed line).

Figure 10

Stream function of the fundamental mode $(n,k,p)=(1,0,0)$ for free b.c.: contour plot of ${\psi }_{1,0,0}^f$ in the middepth plane $(x,y,0)$. Lines correspond to increased level of ${\psi }^f$ of amplitudes $i/8$, $1\le i\le 8$.

Figure 11

Stream function of the fundamental mode $(n,k,p)=(1,0,0)$ for rigid b.c. on the $y$ axis: contour plot of ${\psi }_{1,0,0}^r$ in the middepth plane $(x,y,0)$. Lines correspond to increased level of ${\psi }^r$ of amplitudes $i/8$, $1\le i\le 8$. (a) $L/d=20/2=10$; (b) $L/d=20/4=5$; (c) $L/d=20/6=10/3$. Notice the streamlines similarity between (a) (rigid b.c. and small aspect ratio) and Fig. 10 (free b.c.), despite the different b.c. This contrasts with the increasing elliptic form of streamlines as $L/d$ decreases from (b) to (c).

Figure 12

Propagation mapping starting from a single burnt pixel $P$ at time $t_0$. To approach isotropy on the square lattice, propagation on axes (mapping $M_a$) and on diagonals (mapping $M_d$) are alternated at each time steps $t_i=t_0+i\mathrm{\Delta }t$. Mappings at $t_1\text{:}{M}_{a}P$; at $t_2\text{:}{M}_d\otimes M_{a}P$; at $t_3\text{:}{M}_a\otimes M_d\otimes M_{a}P$.

Figure 13

Simulation of front propagation for free or rigid b.c. Burnt (resp. fresh) medium appears in white [resp. in blue (black)]. Columns correspond to free b.c. (left), rigid b.c. with $d=2$ mm, $L=20$ mm (center), and rigid b.c. with $d=6$ mm, $L=20$ mm (right). Lines correspond to reduced vortex intensity $U/V_0=20$ (above) and $U/V_0=60$ (below). The differences are tiny and mainly concern the distance between the front and the lateral boundaries. These distances are similar for free b.c. and rigid b.c. with $d=2$ mm, but increase for $d=6$ mm.

Figure 14

Kinematic simulation of front propagation for the fundamental modes (33) and (34). Front velocity enhancement for free and rigid b.c. (a) Free b.c.: Full points correspond to the present kinematic simulation of front propagation. The full line interpolates data obtained from simulation of advection-reaction-diffusion dynamics for the same flow and the same b.c. [12]. (b) Rigid b.c.: The full line refers to the fit of the data obtained in panel (a) for free b.c. It is reported for comparison.

Figure 15

Implication of resolution on the kinematic simulation of mean front propagation for free b.c.

Figure 16

Comparison between kinematic numerical simulation and experiment for free b.c. (a) and rigid b.c. (b). (a) Free b.c.: the kinematic simulation is independent of the channel depth. The full line interpolates its data. (b) Rigid b.c.: the kinematic simulation depends on the channel depth. The thin lines interpolate its data for $d=2\text{mm}$ (full line), $d=4\text{mm}$ (dashed line), $d=6\text{mm}$ (dotted line).