Figure 1

Boosted spiral and target patterns of an excitable chemical Belousov-Zhabotinsky reaction in a turbulent flow. Grey scale indicates concentration of ferriin, $\mathrm{Fe}(\mathrm{phen}{)}_3^{3+}$. Upper panel, 1–3: Image sequence of boosted spiral with $\Delta t\approx 3.6\mathrm{s}$, $f=70\mathrm{Hz}$, $a_0\approx 1.8g$, $[{\mathrm{H}}_2{\mathrm{SO}}_4]=1.2\mathrm{M}$. 2nd image: Trajectory of spiral tip in time (color code: early position blue, late position red). Lower panel, 1–3: Image sequence of boosted target wave with $\Delta t\approx 12.4\mathrm{s}$, $f=50\mathrm{Hz}$, $a_0\approx 1.2g$, $[{\mathrm{H}}_2{\mathrm{SO}}_4]=0.6\mathrm{M}$. Both patterns form spontaneously and are persistent phenomena that can last from a few minutes up to one hour. For corresponding movies ($M1$, $M2$) see Supplemental Material [16]. Right: Three close-ups show molecular-diffusion-induced spiral and target patterns in absence of fluid flow in the same container. Note the large difference in scales between these usual and the boosted patterns.Reuse & Permissions

Figure 2

Front velocity of reactive waves in dependence of turbulent diffusion. (a) The velocity of the boosted target wave fronts $v_f$ (crosses) follows the FKPP prediction $v_f=2\sqrt{D_{*}/{\tau }_{\mathrm{reac}}}$ (solid line) derived from the molecular case (circle). Dashed lines indicate the error bounds estimated from the standard deviation of the velocity measurements from the molecular-diffusion-induced target wave. Inset (b) shows a close up of the turbulent data pairs. (c) Target front velocity $v_f$ vs turbulent root-mean-square velocity of the turbulent flow in one direction $v^{\prime }=v_{\mathrm{rms}}/\sqrt{2}$, both normalized to the front velocity $v_{\mathrm{mol}}$ of the molecular-diffusion-induced target wave. (d) The measured diffusion coefficients are shown as a function of the Reynolds number $\mathrm{Re}=v_{\mathrm{rms}}{\lambda }_F/\nu$ indicating the turbulence strength, where $\nu$ is the kinematic viscosity of the fluid. Inset (e) shows the absolute diffusion for the flows with $\mathrm{Re}\approx 43$, $\mathrm{Re}\approx 120$, and $\mathrm{Re}\approx 194$ and the linear fit for estimation of the turbulent diffusion coefficient. Inset (f) shows an exemplary energy spectrum of the flow for $\mathrm{Re}\approx 120$. A double cascade and a regime with a Kolmogorov type scaling ($E_k\propto k^{-5/3}$) can be distinguished. $k_F$ is the typical Faraday wave number.Reuse & Permissions

Figure 3

Front characteristics of boosted target waves. (a), (b) Space-time plots of boosted targets for $D_{*}\approx 5.4{\mathrm{mm}}^2/\mathrm{s}$ $a\approx 1.3g_0$) and $D_{*}\approx 30.0{\mathrm{mm}}^2/\mathrm{s}$ $a\approx 2.2g_0$). Arrows indicate the direction of front propagation (Supplemental Material movies $M2$ and $M8$ [16]). The target waves are narrower, slower, and more filamentous for the smaller diffusion coefficient. (c), (d) Ferriin concentration, [$\mathrm{Fe}(\mathrm{phen}{)}_3^{3+}$], along a line at three different instances of time, $\Delta t\approx 6.4\mathrm{s}$, for $D_{*}\approx 5.4{\mathrm{mm}}^2/\mathrm{s}$ and $D_{*}\approx 30.0{\mathrm{mm}}^2/\mathrm{s}$ respectively. (e), (f) Ferriin concentrations in time for the same values of $D_{*}$ at three different points in space. Time scales of the forward reaction ${\tau }_1$ and the backward reaction ${\tau }_2$ can be estimated as the times of rise and fall of the ferriin concentration. (g) The mean profile of the target waves for both diffusion coefficients estimated by averaging over all targets measured. (h) Different widths $w_1$ and $w_2$ of the profile in dependence of the diffusion coefficient $D_{*}$. The full width $w_2$ of the target wave grows with $\sqrt{D_{*}}$ as expected while the width of the rising edge $w_1$ stays constant.Reuse & Permissions