Planar isotropy of passive scalar turbulent mixing with a mean perpendicular gradient

A recently proposed evolution equation [Vaienti et al., Physica D 85, 405 (1994)] for the probability density functions (PDF's) of turbulent passive scalar increments obtained under the assumptions of fully three-dimensional homogeneity and isotropy is submitted to validation using direct numerical simulation (DNS) results of the mixing of a passive scalar with a nonzero mean gradient by a homogeneous and isotropic turbulent velocity field. It is shown that this approach leads to a quantitatively correct balance between the different terms of the equation, in a plane perpendicular to the mean gradient, at small scales and at large Péclet number. A weaker assumption of homogeneity and isotropy restricted to the plane normal to the mean gradient is then considered to derive an equation describing the evolution of the PDF's as a function of the spatial scale and the scalar increments. A very good agreement between the theory and the DNS data is obtained at all scales. As a particular case of the theory, we derive a generalized form for the well-known Yaglom equation (the isotropic relation between the second-order moments for temperature increments and the third-order velocity-temperature mixed moments). This approach allows us to determine quantitatively how the integral scale properties influence the properties of mixing throughout the whole range of scales. In the simple configuration considered here, the PDF's of the scalar increments perpendicular to the mean gradient can be theoretically described once the sources of inhomogeneity and anisotropy at large scales are correctly taken into account.