Scalar gradients in stirred mixtures and the deconstruction of random fields

A general theory for predicting the distribution of scalar gradients (or concentration differences) in heterogeneous flows is proposed. The evolution of scalar fields is quantified from the analysis of the evolution of elementary lamellar structures, which naturally form under the stretching action of the flows. Spatial correlations in scalar fields, and concentration gradients, hence develop through diffusive aggregation of stretched lamellae. Concentration levels at neighbouring spatial locations result from a history of lamella aggregation, which is partly common to the two locations. Concentration differences eliminate this common part, and thus depend only on lamellae that have aggregated independently. Using this principle, we propose a theory which envisions concentration increments as the result of a deconstruction of the basic lamella assemblage. This framework provides analytical expressions for concentration increment probability density functions (PDFs) over any spatial increments for a range of flow systems, including turbulent flows and low-Reynolds-number porous media flows, for confined and dispersing mixtures. Through this deconstruction principle, scalar increment distributions reveal the elementary stretching and aggregation mechanisms building scalar fields.