An Improved Sparse Direct Interaction Perturbation Closure for Scalar Mixing

New equations that govern the scalar power spectrum of constant-density homogeneous isotropic turbulence have been derived by applying the Sparse Direct-Interaction Perturbation (SDIP) technique. Traditional SDIP treats integrable singularities, which appear at zero time separation, as invariant to arbitrary time separation. This results in a two-time scalar correlation, and response function, that does not incorporate turbulence stirring effects. The proposed higher-order closure relaxes this simplification and allows a more complex coupling between the Lagrangian map and the scalar fields. Fourier analysis of the newly derived equations is required to identify how the traditional solution is modified, and how a previously neglected component evolves to become materially significant. The condition of statistical isotropy simplifies this analysis substantially. The resulting equations demonstrate that a delicate cancellation of singularities must occur for the integrals to converge. The theoretically predicted Obukhov-Corrsin constant from the present closure is in better agreement with experimentally obtained values when compared to the traditional SDIP approach.