Velocity Gradients Along Particle Trajectories in Turbulence
Johnson, Perry L
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The creation of large spatial gradients in velocity by turbulent flows has important implications for a number of micro-physical applications that are sensitive to the straining and rotating influences of the immediate fluid environment. Because velocity gradients tend to be dominated by contributions from the smallest scales of motion in turbulence, their statistics enjoy many similarities across a wide range of natural and man-made flows and the canonical case of isotropic turbulence provides a simple flow in which to explore this aspect of turbulence in detail. In this thesis, the dynamics and kinematics of turbulent velocity gradients experienced while following Lagrangian trajectories are explored using fully resolved simulations and new, accurate techniques for inexpensive, reduced order models are developed. In particular, the cumulative stretching of infinitesimal material volumes is quantified statistically using large-deviation theory and compared with the stretching of vorticity. Following this, the dynamics of the velocity gradient itself are modeled using a stochastic approach. While some important terms are represented exactly in the Lagrangian formulation of velocity gradient dynamics, closure approximations are constructed systematically by applying a Lagrangian deformation map to Gaussian field statistics. This model is then extended to arbitrarily high Reynolds numbers using a multiple time scale expansion which faithfully represents energy cascade dynamics and the broad range of timescales present in high Reynolds number flows. It is also demonstrated that this stochastic modeling approach provides an accurate, an inexpensive way to model velocity gradients in coarse-grained simulations of inhomogeneous flows where the small scales of turbulence are not resolved. Finally, the restricted Euler model for Lagrangian velocity gradients is extended to inertial particle trajectories. While the model inherits the restricted Euler finite time singularity, qualitative features of velocity gradients on inertial particle trajectories are correctly predicted. Results point to the possibility for future developments of higher-fidelity models for applications where particle density differs significantly from that of the surrounding fluid.