Relative motion of two neighboring points on inert and reactive scalar iso-surfaces in homogeneous turbulence

Abstract

The study of the deformation and rotation of line and surface and volume elements, embedded in a mixing-reaction zone, is supposedly of a paramount importance to comprehend the evolution of inert and chemically reacting species in turbulent flows. This analysis examines the relative motion of two points on adjacent, nonmaterial, propagating, isoscalar surfaces, subject to the flow velocity, v, and the normal displacement speed vector, Sdn (n is the unit vector perpendicular to the isosurface). The methodology of Perry and Chong [“Topology of flow patterns in vortex motions and turbulence,” Appl. Sci. Res. 53, 357–374 (1994)], applied to the velocity gradient tensors of v, A, and Sdn, Aa, serves to characterize this motion. Small-scale flow and displacement topologies clearly emerge from this description. The invariants of these two velocity gradient tensors yield the total rate of change of the normal distance between two isosurfaces, the area stretch factor, the total volumetric dilatation rate, and the nodal or focal features of the motions. The influence of the first invariant on the displacement speed and the chemical depletion is also established. A simple mathematical formalism portrays the time rate of change of an infinitesimal nonmaterial vector, r, joining two points on adjacent isoscalar surfaces within a zone where turbulent mixing of inert and reactive species occurs. Databases of a 3D 5123 DNS of a statistically homogeneous and stationary turbulence with inert and reactive scalars undergoing random mixing in a constant density fluid are examined to illustrate the application of the previous conceptual framework. Small scales are well resolved. Various flow and displacement contributions to the two-point relative velocity components, normal and tangential to the isosurface, are compared to conclude that motions of isoscalar surfaces due to Sdn are, at least, as important as those caused by v. In particular, the additional vorticity, ωa, is one order of magnitude greater than the flow vorticity, ω, and tangential to the isoscalar surfaces, contributing significantly to their folding. While the dynamically passive scalars do not modify the conventional local flow motions, new additional topologies, induced by Aa and typified by its invariants, Pa, Qa, and Ra, appear and affect the displacement speed and the reaction rate. The mass entrainment rate per unit mass into a volume element between two adjacent isosurfaces is given by the first invariant, −Pa, and its influence on the displacement speed and the chemical reaction is explored.