Stochastic Shell Model for Turbulent Mixing of Multiple Scalars with Mean Gradients and Differential Diffusion

In this paper, we develop a shell model for the velocity and scalar concentrations that, by design, is consistent with the eddy damped quasi-normal Markovian (EDQNM) model for multiple mixing scalars. We review the realizable form of the EDQNM model derived by Ulitsky and Collins (J Fluid Mech 412:303–329, 2000 ), which forms the basis for the shell model. The equations governing the velocity and scalar within each shell are stochastic ordinary differential equations with drift and diffusion terms chosen so that the velocity variance, velocity–scalar cross correlations, and scalar–scalar cross correlations within each shell precisely match the EDQNM model predictions. Consequently, shell averages can be thought of as a representation of the discrete three-dimensional spectrum. An advantage the shell model has over the original EDQNM equations is that the sum of each realization over the shells is a model for the fine-grained, joint velocity/scalar probability density function (PDF). Indeed, this provides some of the motivation for the development of the model. We cannot exploit this feature in the present study of the mixing of two scalars with uniform mean gradients, as the PDF is a joint Gaussian throughout (and hence the correlation matrix completely defines the distribution). The model is capable of predicting Lagrangian correlation functions for the scalar, scalar dissipation and velocity. We find the predictions of the model are in good qualitative agreement with direct numerical simulations by Yeung (J Fluid Mech 427:241–274, 2001 ). Eventually we will apply the shell model to scalars that are initially highly non-Gaussian (e.g., double delta function) and observe the relaxation towards a Gaussian. As the shell model contains information on the spectral distribution of the scalar field, the relaxation rate will depend upon the length and time scales of the turbulence and the scalar fields, as well as the molecular diffusivities of the species. The full capabilities of the PDF predictions of the model will be the subject of a future publication.