6 Reactive medium in turbulent flow

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.

Figure 13:
Density of component (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). |

Figure 14:
Density of components and
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.

Figure 15:
Probability density function of 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 as a function of size (see Fig. 16) using the same procedure as for Fig. 9. The slope is (instead of 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.

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