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This study develops a comprehensive description of local streamline geometryand uses the resulting shape features to characterize velocity gradient($A_{ij}$) dynamics. The local streamline geometric shape parameters andscale-factor (size) are extracted from $A_{ij}$ by extending the linearizedcritical point analysis. In the present analysis, $A_{ij}$ is factorized intoits 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 byfour $b_{ij}$ parameters -- second invariant, $q$; third invariant, $r$;intermediate strain-rate eigenvalue, $a_2$; and, angle between vorticity andintermediate strain-rate eigenvector, $\omega_2$. Velocity gradient magnitude$A$ plays a role only in determining the scale of the local streamlinestructure. Direct numerical simulation data of forced isotropic turbulence($Re_\lambda \sim 200 - 600$) is used to establish streamline shape and scaledistribution and, then to characterize velocity-gradient dynamics. Conditionalmean trajectories (CMTs) in $q$-$r$ space reveal important non-local featuresof pressure and viscous dynamics which are not evident from the$A_{ij}$-invariants. Two distinct types of $q$-$r$ CMTs demarcated by aseparatrix are identified. The inner trajectories are dominated byinertia-pressure interactions and the viscous effects play a significant roleonly 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 throughdifferent topologies correlate well with the corresponding populationfractions. These findings not only lead to improved understanding of non-localdynamics, but also provide an important foundation for developing Lagrangianvelocity-gradient models.

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