Exact transport equation for eddy diffusivity in turbulent shear flow

Two-equation models that treat the transport equations for two variables are typical models for the Reynolds-averaged Navier–Stokes equation. Compared to the equation for the turbulent kinetic energy, the equation for the second variable such as the dissipation rate does not have a theoretical analogue. In this work, the exact transport equation for the eddy diffusivity was derived and examined for better understanding turbulence and improving two-equation models. A new length scale was first introduced, which involves the response function for the scalar fluctuation. It was shown that the eddy diffusivity can be expressed as the correlation between the velocity fluctuation and the new length scale. The transport equations for the eddy diffusivity and the length-scale variance were derived theoretically. Statistics such as terms in the transport equations were evaluated using the direct numerical simulation of turbulent channel flow. It was shown that the streamwise component of the eddy diffusivity is greater than the other two components in the whole region. In the transport equation for the eddy diffusivity, the production term due to the Reynolds stress is a main positive term, whereas the pressure–length-gradient correlation term plays a role of destruction. It is expected that the analysis of the transport equations is helpful in developing better turbulence models.