Stretching, folding, and alignment along Lagrangian orbits

Abstract

The stirring and mixing of passive scalars is determined by the geometry of the Lagrangian orbits followed by the advected particles. The efficiency of these processes depends on the ‘‘stretching and folding’’ of a local element (line, area, and volume). This deformation is determined by the nature of the local straining field and the degree of alignment of the advected element with the straining directions. Various ways of quantifying the dynamics of the straining field, that are valid for both three-dimensional flows and dynamical systems in general, are proposed. One of the simplest is a quantity termed the persistence of strain, σ2, which is defined as σ2= 1/2 Tr(A2), where A is the tangent map of the flow. This is shown to provide a compact quantification of the relative strengths of the straining and rotational components of the flow along a Lagrangian path. This, in turn, leads to the concept of a ‘‘strain history’’ of a fluid particle for which we can also compute a ‘‘stretch–fold’’ radio χ, that provides an average measure of these effects along an orbit. The quantity σ2 is closely related to the curvature of an orbit and this suggests a study of the curvature κ and torsion τ of particle trajectories. Further information is provided by extending the definition of Lyapunov exponents ins such a way as to include complex eigenvalue information that is traditionally discarded from the tangent map during the computation of these exponents. This provides additional information about the orientation and alignment of the advected element. These considerations are especially important when considering the fate of small deformable bodies, such as polymers, in flow fields with rapidly fluctuating strain rates.