Since the availability of data from direct numerical simulation
(DNS) of turbulence,
researchers have utilized the joint PDFs of invariants of the velocity
gradient tensor
to study the geometry of small-scale motions of turbulence. However, the
joint PDFs
only give an instantaneous static representation of the properties of fluid
particles and
dynamical Lagrangian information cannot be extracted. In this paper, the
Lagrangian
evolution of the invariants of the velocity gradient tensor is studied
using conditional
mean trajectories (CMT). These CMT are derived using the concept of the
conditional
mean time rate of change of invariants calculated from a numerical simulation
of
isotropic turbulence. The study of the CMT in the invariant space
(RA, QA) of
the velocity-gradient tensor, invariant space
(RS, QS) of
the rate-of-strain tensor, and invariant space
(RW, QW) of
the rate-of-rotation tensor show that the mean evolution in the
(Σ, QW) phase plane,
where Σ is the vortex stretching, is cyclic with a
characteristic period similar to that found by Martin et al. (1998)
in the cyclic mean
evolution of the CMT in the (RA, QA)
phase plane. Conditional mean trajectories in
the (Σ, QW) phase plane suggest
that the initial
reduction of QW in regions of high
QW is due to viscous diffusion and that vorticity
contraction only plays a secondary
role subsequent to this initial decay. It is also found that in regions
of the flow with
small values of QW, the local values of
QW do not begin to increase, even in the
presence of self-stretching, until a certain self-stretching rate threshold
is reached, i.e.
when Σ≈0.25 〈QW〉1/2.
This study also shows that in regions where the kinematic
vorticity number (as defined by Truesdell 1954) is low, the local value
of dissipation
tends to increase in the mean as observed from a Lagrangian frame of reference.
However, in regions where the kinematic vorticity number is high, the local
value of
enstrophy tends to decrease. From the CMT in the
(−QS, RS
phase plane,
it is also deduced that for large values of dissipation, there is a tendency
for fluid particles
to evolve towards having a positive local value of the intermediate principal
rate of
strain.
Dynamics and invariants of the perceived velocity gradient tensor in homogeneous and isotropic turbulence