  We study turbulent diffusion of chemically reacting gaseous admixtures in a
developed turbulence. In our previous study [Phys. Rev. Lett. {\bf 80}, 69
(1998)] using a path-integral approach for a delta-correlated in time random
velocity field, we demonstrated a strong modification of turbulent transport in
fluid flows with chemical reactions or phase transitions. In the present study
we use the spectral tau approximation, that is valid for large Reynolds and
Peclet numbers, and show that turbulent diffusion of the reacting species can
be strongly depleted by a large factor that is the ratio of turbulent and
chemical times (turbulent Damk\"{o}hler number). We have demonstrated that the
derived theoretical dependence of turbulent diffusion coefficient versus the
turbulent Damk\"{o}hler number is in a good agreement with that obtained
previously in the numerical modelling of a reactive front propagating in a
turbulent flow and described by the Kolmogorov-Petrovskii-Piskunov-Fisher
equation. We have found that turbulent cross-effects, e.g., turbulent mutual
diffusion of gaseous admixtures and turbulent Dufour-effect of the chemically
reacting gaseous admixtures, are less sensitive to the values of stoichiometric
coefficients. The mechanisms of the turbulent cross-effects are different from
the molecular cross effects known in irreversible thermodynamics. In a fully
developed turbulence and at large Peclet numbers the turbulent cross-effects
are much larger than the molecular ones. The obtained results are applicable
also to heterogeneous phase transitions.
