Multi-scale model of gradient evolution in turbulent flows

Abstract

A multi-scale model for the evolution of the velocity gradient tensor in fully developed turbulence is proposed. The model is based on a coupling between a ``Restricted Euler'' dynamics [{\it P. Vieillefosse, Physica A, {\bf 14}, 150 (1984)}] which describes gradient self-stretching, and a deterministic cascade model which allows for energy exchange between different scales. We show that inclusion of the cascade process is sufficient to regularize the well-known finite time singularity of the Restricted Euler dynamics. At the same time, the model retains topological and geometrical features of real turbulent flows: these include the alignment between vorticity and the intermediate eigenvector of the strain-rate tensor and the typical teardrop shape of the joint probability density between the two invariants, $R-Q$, of the gradient tensor. The model also possesses skewed, non-Gaussian longitudinal gradient fluctuations and the correct scaling of energy dissipation as a function of Reynolds number. Derivative flatness coefficients are in good agreement with experimental data.

4 pages, 4 figures

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• Lagrangian Refined Kolmogorov Similarity Hypothesis for Gradient Time-evolution in Turbulent Flows
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• Recent Fluid Deformation closure for velocity gradient tensor dynamics in turbulence: time-scale effects and expansions
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• Exploring the turbulent velocity gradients at different scales from the perspective of the strain-rate eigenframe
• Restricted Euler dynamics along trajectories of small inertial particles in turbulence
• The turbulence velocity gradient tensor formed additively by normal and non-normal tensors
• Restricted Euler dynamics in free-surface turbulence
• Scaling of conditional Lagrangian time correlation functions of velocity and pressure gradient magnitudes in isotropic turbulence
• Physics-Informed Machine Learning of the Lagrangian Dynamics of Velocity Gradient Tensor