Characterization of velocity-gradient dynamics in incompressible turbulence using local streamline geometry

Abstract

This study develops a comprehensive description of local streamline geometry and uses the resulting shape features to characterize velocity gradient ($A_{ij}$) dynamics. The local streamline geometric shape parameters and scale-factor (size) are extracted from $A_{ij}$ by extending the linearized critical point analysis. In the present analysis, $A_{ij}$ is factorized into its magnitude ($A \equiv \sqrt{A_{ij}A_{ij}}$) and normalized tensor $b_{ij} \equiv A_{ij}/A$. The geometric shape is shown to be determined exclusively by four $b_{ij}$ parameters -- second invariant, $q$; third invariant, $r$; intermediate strain-rate eigenvalue, $a_2$; and, angle between vorticity and intermediate strain-rate eigenvector, $ω_2$. Velocity gradient magnitude $A$ plays a role only in determining the scale of the local streamline structure. Direct numerical simulation data of forced isotropic turbulence ($Re_λ\sim 200 - 600$) is used to establish streamline shape and scale distribution and, then to characterize velocity-gradient dynamics. Conditional mean trajectories (CMTs) in $q$-$r$ space reveal important non-local features of pressure and viscous dynamics which are not evident from the $A_{ij}$-invariants. Two distinct types of $q$-$r$ CMTs demarcated by a separatrix are identified. The inner trajectories are dominated by inertia-pressure interactions and the viscous effects play a significant role only in the outer trajectories. Dynamical system characterization of inertial, pressure and viscous effects in the $q$-$r$ phase space is developed. Additionally, it is shown that the residence time of $q$-$r$ CMTs through different topologies correlate well with the corresponding population fractions. These findings not only lead to improved understanding of non-local dynamics, but also provide an important foundation for developing Lagrangian velocity-gradient models.