  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.
