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

• Published in: Journal of fluid mechanics Vol. 895
• Main Authors: Das, Rishita, Girimaji, Sharath S.
• Format: Journal Article
• Language: English
• Published: Cambridge Cambridge University Press 25.07.2020
• Subjects: Classification; Computational fluid dynamics; Computer simulation; Critical point; Direct numerical simulation; Dynamics; Eigenvalues; Eigenvectors; Geometry; Inertia; Invariants; Isotropic turbulence; Mathematical models; Parameters; Pressure; Pressure effects; Ratios; Residence time; Shape; Strain rate; Streamlines; Tensors; Topology; Turbulence; Velocity; Velocity gradient; Velocity gradients; Vorticity
• ISSN: 0022-1120; 1469-7645
• DOI: 10.1017/jfm.2020.286

This study develops a comprehensive description of local streamline geometry and uses the resulting shape features to characterize velocity gradient ( $\unicode[STIX]{x1D608}_{ij}=\unicode[STIX]{x2202}u_{i}/\unicode[STIX]{x2202}x_{j}$ ) dynamics. The local streamline geometric shape parameters and scale factor (size) are extracted from $\unicode[STIX]{x1D608}_{ij}$ by extending the linearized critical point analysis. In the present analysis, $\unicode[STIX]{x1D608}_{ij}$ is factorized into its magnitude ( $A\equiv \sqrt{\unicode[STIX]{x1D608}_{ij}\unicode[STIX]{x1D608}_{ij}}$ ) and normalized tensor $\unicode[STIX]{x1D623}_{ij}\equiv \unicode[STIX]{x1D608}_{ij}/A$ . The geometric shape is shown to be determined exclusively by four $\unicode[STIX]{x1D623}_{ij}$ parameters: second invariant, $q$ ( $=Q/A^{2}$ ); third invariant, $r$ ( $=R/A^{3}$ ); intermediate strain rate eigenvalue, $a_{2}$ ; and vorticity component along intermediate strain rate eigenvector, $\unicode[STIX]{x1D714}_{2}$ . 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 ( $Re_{\unicode[STIX]{x1D706}}\sim 200{-}600$ ) 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 $\unicode[STIX]{x1D608}_{ij}$ -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.