Invariants of the velocity-gradient, rate-of-strain, and rate-of-rotation tensors across the turbulent/nonturbulent interface in jets

Abstract

The invariants of the velocity gradient (R and Q), rate-of-strain (RS and QS), and rate-of-rotation (QW) tensors are analyzed across the turbulent/nonturbulent (T/NT) interface by using a direct numerical simulation (DNS) of a turbulent plane jet at Reλ≈120. The invariants allow a detailed characterization of the dynamics, geometry and topology of the flow during the entrainment. The invariants Q and QS are almost equal and negative outside the turbulent region close to the T/NT interface, which shows the existence of high values of strain product (hence viscous dissipation of kinetic energy) at that location. Right at the T/NT interface, the invariants QW and QS show that virtually all flow points there are characterized by irrotational dissipation, with no discernible sign of the coherent structures which are known to exist deep inside the turbulent region. Moreover, the invariants of the velocity gradient tensor (Q and R) show that the classical “teardrop” shape of their associated phase map is not yet formed at the T/NT interface. All the invariants rapidly change after the T/NT interface is crossed into the turbulent region. For instance, the enstrophy density, proportional to QW, is zero in the irrotational flow region and high and more or less constant inside the turbulent region, after it undergoes a sharp jump near the T/NT interface. Inside the turbulent region, at a distance of only 1.7η from the T/NT interface, where η is the Kolmogorov microscale, the invariants QW and QS suggest that large scale coherent vortices already exist in the flow. Furthermore, the joint probability density function of Q and R already displays its well known teardrop shape at that location. Moreover, the geometry of the straining (or deformation) of the fluid elements during the turbulent entrainment process is preferentially characterized by biaxial expansion with αS:βS:γS=2:1:−3, where αS, βS, and γS are the eigenvalues of the rate-of-strain tensor arranged in descending order. Based on an analysis of the invariants, many aspects of the flow topology inside the turbulent region at a distance of only 1.7η from the T/NT interface are already similar to those observed deep inside the turbulent region.