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Velocity gradient tensor, $A_{ij}\equiv \partial u_i/\partial x_j$ , in a turbulence flow field is modelled by separating the treatment of intermittent magnitude ($A = \sqrt {A_{ij}A_{ij}}$) from that of the more universal normalised velocity gradient tensor, $b_{ij} \equiv A_{ij}/A$. The boundedness and compactness of the $b_{ij}$ -space along with its universal dynamics allow for the development of models that are reasonably insensitive to Reynolds number. The near-lognormality of the magnitude $A$ is then exploited to derive a model based on a modified Ornstein–Uhlenbeck process. These models are developed using data-driven strategies employing high-fidelity forced isotropic turbulence data sets. A posteriori model results agree well with direct numerical simulation data over a wide range of velocity-gradient features and Reynolds numbers. [ABSTRACT FROM AUTHOR]

Fish often swim in crystallized group formations (schooling) and orient themselves against the incoming flow (rheotaxis). At the intersection of these two phenomena, we investigate the emergence of unique schooling patterns through passive hydrodynamic mechanisms in a fish pair, the simplest subsystem of a school. First, we develop a fluid dynamics-based mathematical model for the positions and orientations of two fish swimming against a flow in an infinite channel, modelling the effect of the self-propelling motion of each fish as a point-dipole. The resulting system of equations is studied to gain an understanding of the properties of the dynamical system, its equilibria and their stability. The system is found to have five types of equilibria, out of which only upstream swimming in in-line and staggered formations can be stable. A stable near-wall configuration is observed only in limiting cases. It is shown that the stability of these equilibria depends on the flow curvature and streamwise interfish distance, below critical values of which, the system may not have a stable equilibrium. The study reveals that simply through passive fluid dynamics, in the absence of any other feedback/sensing, we can justify rheotaxis and the existence of stable in-line and staggered schooling configurations. [ABSTRACT FROM AUTHOR]

The local streamline topology classification method of Chong et al. (Phys. Fluids A : Fluid Dyn., vol. 2, no. 5, 1990, pp. 765–777) is adapted and extended to describe the geometry of infinitesimal vortex lines. Direct numerical simulation (DNS) data of forced isotropic turbulence reveals that the joint probability density function (p.d.f.) of the second ($q_\omega$) and third ($r_\omega$) normalized invariants of the vorticity gradient tensor asymptotes to a self-similar bell shape for $Re_\lambda > 200$. The same p.d.f. shape is also seen at the late stages of breakdown of a Taylor–Green vortex suggesting the universality of the bell-shaped p.d.f. form in turbulent flows. Additionally, vortex reconnection from different initial configurations is examined. The local topology and geometry of the reconnection bridge is shown to be nearly identical in all cases considered in this work. Overall, topological characterization of the vorticity field provides a useful analytical basis for examining vorticity dynamics in turbulence and other fluid flows. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Turbulence small-scale behavior has been commonly investigated in literature by decomposing the velocity-gradient tensor (Aij) into the symmetric strain-rate (Sij) and anti-symmetric rotation-rate (Wij) tensors. To develop further insight, we revisit some of the key studies using a triple decomposition of the velocity-gradient tensor. The additive triple decomposition formally segregates the contributions of normal-strain-rate (Nij), pure-shear (Hij) and rigid-body-rotation-rate (Rij). The decomposition not only highlights the key role of shear, but it also provides a more accurate account of the influence of normal-strain and pure rotation on important small-scale features. First, the local streamline topology and geometry are described in terms of the three constituent tensors in velocity-gradient invariants' space. Using direct numerical simulation (DNS) data sets of forced isotropic turbulence, the velocity-gradient and pressure field fluctuations are examined at different Reynolds numbers. At all Reynolds numbers, shear contributes the most and rigid-body-rotation the least toward the velocity-gradient magnitude (A2 ≡ AijAij). Especially, shear contribution is dominant in regions of high values of A2 (intermittency). It is shown that the high-degree of enstrophy intermittency reported in literature is due to the shear contribution toward vorticity rather than that of rigid-body-rotation. The study also provides an explanation for the non-intermittent nature of pressure-Laplacian, despite the strong intermittency of enstrophy and dissipation fields. The study further investigates the alignment of the rotation axis with normal strain-rate and pressure Hessian eigenvectors. Overall, it is demonstrated that triple decomposition offers unique and deeper understanding of velocity-gradient behavior in turbulence. [ABSTRACT FROM AUTHOR]

We seek to examine the changes in velocity-gradient structure (local streamline topology) and related dynamics as a function of Reynolds number ($Re_{\unicode[STIX]{x1D706}}$). The analysis factorizes the velocity gradient ($\unicode[STIX]{x1D608}_{ij}$) into the magnitude ($A^{2}$) and normalized-gradient tensor ($\unicode[STIX]{x1D623}_{ij}\equiv \unicode[STIX]{x1D608}_{ij}/\sqrt{A^{2}}$). The focus is on bounded $\unicode[STIX]{x1D623}_{ij}$ as (i) it describes small-scale structure and local streamline topology, and (ii) its dynamics is shown to determine magnitude evolution. Using direct numerical simulation (DNS) data, the moments and probability distributions of $\unicode[STIX]{x1D623}_{ij}$ and its scalar invariants are shown to attain $Re_{\unicode[STIX]{x1D706}}$ independence. The critical values beyond which each feature attains $Re_{\unicode[STIX]{x1D706}}$ independence are established. We proceed to characterize the $Re_{\unicode[STIX]{x1D706}}$ dependence of $\unicode[STIX]{x1D623}_{ij}$ -conditioned statistics of key non-local pressure and viscous processes. Overall, the analysis provides further insight into velocity-gradient dynamics and offers an alternative framework for investigating intermittency, multifractal behaviour and for developing closure models. [ABSTRACT FROM AUTHOR]