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Abstract

Understanding the non-local pressure contributions and viscous effects
on the small-scale statistics remains one of the central challenges in
the study of homogeneous isotropic turbulence. Here we address this
issue by studying the impact of the pressure Hessian as well as viscous
diffusion on the statistics of the velocity gradient tensor in the
framework of an exact statistical evolution equation. This evolution
equation shares similarities with earlier phenomenological models for
the Lagrangian velocity gradient tensor evolution, yet constitutes the
starting point for a systematic study of the unclosed pressure Hessian
and viscous diffusion terms. Based on the assumption of incompressible
Gaussian velocity fields, closed expressions are obtained as the results
of an evaluation of the characteristic functionals. The benefits and
shortcomings of this Gaussian closure are discussed, and a
generalization is proposed based on results from direct numerical
simulations. This enhanced Gaussian closure yields, for example,
insights on how the pressure Hessian prevents the finite-time
singularity induced by the local self-amplification and how its
interaction with viscous effects leads to the characteristic strain
skewness phenomenon.