Recent Fluid Deformation closure for velocity gradient tensor dynamics in turbulence: time-scale effects and expansions

TL;DR

This paper investigates the limitations of the Recent Fluid Deformation closure in turbulence modeling, analyzing how time-scale choices and stochastic forcing affect the accuracy and stability of velocity gradient tensor predictions.

Contribution

The study provides a detailed analysis of the closure's limitations, explores the effects of time-correlated forcing, and introduces a linearized model for stability insights.

Findings

Reducing decorrelation time scale does not proportionally reduce autocorrelation time.

Inconsistencies in time-scale choices may cause unphysical predictions at high Reynolds numbers.

Linearized model offers stability insights but does not extend applicability limits.

Abstract

In order to model pressure and viscous terms in the equation for the Lagrangian dynamics of the velocity gradient tensor in turbulent flows, Chevillard & Meneveau (Phys. Rev. Lett. 97, 174501, 2006) introduced the Recent Fluid Deformation closure. Using matrix exponentials, the closure allows to overcome the unphysical finite-time blow-up of the well-known Restricted Euler model. However, it also requires the specification of a decorrelation time scale of the velocity gradient along the Lagrangian evolution, and when the latter is chosen too short (or, equivalently, the Reynolds number is too high), the model leads to unphysical statistics. In the present paper, we explore the limitations of this closure by means of numerical experiments and analytical considerations. We also study the possible effects of using time-correlated stochastic forcing instead of the previously employed white-noise forcing. Numerical experiments show that reducing the correlation time scale specified in the closure and in the forcing does not lead to a commensurate reduction of the autocorrelation time scale of the predicted evolution of the velocity gradient tensor. This observed inconsistency could explain the unrealistic predictions at increasing Reynolds numbers.We perform a series expansion of the matrix exponentials in powers of the decorrelation time scale, and we compare the full original model with a linearized version. The latter is not able to extend the limits of applicability of the former but allows the model to be cast in terms of a damping term whose sign gives additional information about the stability of the model as function of the second invariant of the velocity gradient tensor.