It is then possible to add the effects of that non-trivial local dynamic to turbulent mixing. We select the model of Sect. 5. In order to study the influence of the turbulence, we again do a comparison between a turbulent and a Gaussian velocity field (Fig. 13 for one example). Note that A, B (chemical species), U (internal energy) and R (proportional to the total density) are advected as described by Eq. (11) extended to 4 variables.

$\begin{figure} \par\includegraphics[width=6.8cm,clip]{MS2137f13a.eps}\par\includegraphics[width=6.8cm,clip]{MS2137f13b.eps} \end{figure}$

Figure 13: Density of component $A\protect$ (see Eq. (12)) in a bistable case. The horizontal axis represents position and time flows from top to bottom, using the same initial conditions. Only the latest iterations are shown. Lowest densities are deep blue and highest densities light red. Top panel: mixing by a Gaussian velocity field; bottom panel: mixing by a turbulent field (see Appendix D for a much longer time evolution).

$\begin{figure} \includegraphics[width=8.8cm,clip]{MS2137f14.eps} \end{figure}$

Figure 14: Density of components $A\protect$ and $R\protect$ as a function of position at a given time (last time step of Fig. 13 i.e. last "line''), for a turbulent field.

We can observe that, as in the previous case (Sect. 4.3), structures appear naturally at all scales in the turbulent case and not with a Gaussian velocity field. Appendix D shows the variations of A on a much larger time scale. Different initial conditions give indistinguishable results, suggesting that steady state is reached. However, we cannot exclude that some long time drift may still exist which our computation is unable to uncover.

Turbulent structures are not the same for the different components (A, B, R) neither in position nor in size, as can be seen in Fig. 14, which is an horizontal cut of Fig. 13. In order to study more precisely these effects, we plot the probability density function of A(Fig. 15) which should be compared to Fig 8.

$\begin{figure} \includegraphics[width=8.8cm,clip]{MS2137f15.eps} \end{figure}$

Figure 15: Probability density function of $A\protect$ for two different velocity fields: a Gaussian random field and a turbulent velocity field.

Note that the presence of a non-uniform velocity field leads to a single-peaked broad distribution. Turbulence leads to a broader distribution and extended wings. Thus the probability to find some regions at far from equilibrium values is enhanced.

We also plotted the density of $\alpha =\sqrt{\frac{k_{3}}{k_{4}}}n_{A}$ as a function of size (see Fig. 16) using the same procedure as for Fig. 9. The slope is $-0.17\pm 0.03$ (instead of $-0.32\pm 0.03$ for the total density, cf. Sect. 4.3). The evolution through scales is therefore significantly different for one particular component and for the total density.

6 Reactive medium in turbulent flow