A theoretical study of the turbulent diffusion in incompressible shear flows and in passive scalars

The triple velocity, the pressure-velocity, and the scalar-scalar-velocity correlations in incompressible turbulent shear flows and in passive scalar fields are studied by using the two-scale direct-interaction approximation, which satisfies the solenoidal condition of the velocity fields and systematically introduces the cutoff wave numbers. As a result, we obtain their expressions of the gradient diffusion type involving the effect of cross-diffusion. The model constants in the obtained expressions for the triple velocity and the pressure - velocity correlations are close to the values used in the turbulence models. The derived expression for the scalar-scalar-velocity correlation shows the complex dependence on the ratio R of the velocity length scale to the scalar one. As a special case of R= 1, we obtain its explicit expression whose model constant also agrees well with the numerically optimized value.