Front propagation in a chaotic flow field

We investigate numerically the dynamics of a propagating front in the presence of a spatiotemporally chaotic flow field. The flow field is the three-dimensional time-dependent state of spiral defect chaos generated by Rayleigh-Bénard convection in a spatially extended domain. Using large-scale parallel numerical simulations, we simultaneously solve the Boussinesq equations and a reaction-advection-diffusion equation with a Fischer-Kolmogorov-Petrovskii-Piskunov reaction for the transport of the scalar species in a large-aspect-ratio cylindrical domain for experimentally accessible conditions. We explore the front dynamics and geometry in the low-Damköhler-number regime, where the effect of the flow field is significant. Our results show that the chaotic flow field enhances the front propagation when compared with a purely cellular flow field. We quantify this enhancement by computing the spreading rate of the reaction products for a range of parameters. We use our results to quantify the complexity of the three-dimensional front geometry for a range of chaotic flow conditions.

Figure 1

Flow field patterns after one horizontal diffusion time ${\tau }_h=1600$ for three different Rayleigh numbers $R$: (a) $R=2000$ $(\epsilon =0.17)$, (b) $R=4000$ $(\epsilon =1.34)$, and (c) $R=20000$ $(\epsilon =10.71)$. Shown are color contours of the temperature field at a horizontal midplane slice where red (light) is hot rising fluid and blue (dark) is cold sinking fluid. These flow fields are used as initial conditions for simulations that include the simultaneous evolution of the reaction-advection-diffusion equation.

Figure 2

Variation of the characteristic convective velocity $U$ with the reduced Rayleigh number $\epsilon$. The dashed line is a linear curve fit through the data where $U=6.0\epsilon +7.5$. Here $U$ is the average of the maximum fluid velocity in the domain and the error bars represent the standard deviation about the mean for a time period of one vertical diffusion time.

Figure 3

Mean-square displacement $M_2\left(t\right)$ as a function of time $t$ for a system without advection $(R=0)$. Curves are shown for $\xi =\{0,1/2,1\}$, with the curves for $\xi =0$ and 1 labeled. At large times $M_2\left(t\right)$ can be described as $M_2\left(t\right)\propto t^{\gamma }$. In the absence of a reaction $(\xi =0)$ the results show that $\gamma =1$ as expected for purely diffusive transport. For $\xi =1/2$ and 1 the front propagates with $\gamma =2$. The dashed curves are included to provide a reference.

Figure 4

Front propagation in a chaotic flow field for $R=4000$ and $\xi =1/2$. These values yield a Damköhler number of $\text{Da}=0.03$. Images are shown for (a) $t=8$, (b) $t=12$, and (c) $t=16$. The color contours represent the concentration of products where red (light) is a large concentration and blue (dark) is a small concentration indicating the presence of reactants. The black contour lines indicate convection roll boundaries.

Figure 5

Vertical structure of propagating fronts for $\xi =1/2$ at time $t=4$ for the different flow field conditions of cellular flow (top) and chaotic flow (bottom). Color contours are shown for the concentration field of products $c$ where red (light) is a large concentration and blue (dark) is a low concentration. Shown on the top is $R=2000$, where the lateral sidewalls are hot to force a target pattern of convection rolls. Shown on the bottom is $R=6000$, where the flow field is spiral defect chaos. Time is measured after the initiation of the reaction at the center of the domain.

Figure 6

Spreading $M_2\left(t\right)$ with time $t$ for a chaotic flow field with $R=6000$ and $\xi =\{0,1/2,1\}$, with the curves for $\xi =\{0,1\}$ labeled. At long times the spreading can be described as $M_2\left(t\right)\propto t^{\gamma }$. For $\xi =0$ there is only advection and diffusion and a curve fit yields $\gamma =1.09\pm 0.03$. The lower dashed curve has a slope of $\gamma =1$ and indicates diffusive transport. For $\xi >0$ there is reaction, advection, and diffusion. For $\xi =1/2$ this yields $\text{Da}=0.02$ and $\gamma =2.53\pm 0.05$. For $\xi =1$ this yields $\text{Da}=0.04$ and $\gamma =2.43\pm 0.05$. For $\xi >0$ the front propagation is enhanced at long times, as indicated by the upper dashed curve with exponent $\gamma =2.5$.

Figure 7

Image of the propagating front at time $t=15$ for $R=6000$ and $\xi =1/2$. For these parameters $\text{Da}=0.02$. The propagating front is shown in the $z=1/2$ plane.

Figure 8

Variation of the area of the front $A_f$ with time $t$ for $R=\{0,4000,6000\}$. For each value of $R$ there are curves for $\xi =\{1/2,1\}$, with the lower curve for $\xi =1/2$. The dashed lines are curve fits through the data of the form $A_f\propto t^{\zeta }$: For $R=0$, $\zeta =1.10\pm 0.03$; $R=4000$, $\zeta =1.70\pm 0.04$; and $R=6000$, $\zeta =1.50\pm 0.04$. The area has been normalized by the area of the domain $A_0=\pi {\Gamma }^2$.

Figure 9

Variation of the effective perimeter $\eta$ with time where $\eta$ is the ratio of the actual perimeter of the front to the perimeter of a circular front of the same averaged radius. The upper curve is for $R=6000$ and $\xi =1$, which yields $\text{Da}=0.04$, and the lower curve is for a reacting and diffusing system where $R=0$ and $\xi =1$.