A study of the evolution and characteristics of the invariants of the velocity-gradient tensor in isotropic turbulence

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 (R-A, Q(A)) of the velocity-gradient tensor, invariant space (Rs,es) of the rate-of-strain tensor, and invariant space (Rw,Q(W)) of the rate-of-rotation tensor show that the mean evolution in the (Sigma, Q(W)) phase plane, where Sigma 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 (Sigma, Q(W)) phase plane suggest that the initial reduction of Q(W) in regions of high Q(W) 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 Q(W), the local values of Q(W) do not begin to increase, even in the presence of self-stretching, until a certain self-stretching rate threshold is reached, i.e. when Sigma approximate to 0.25 [Q(W)](1/2). This study also shows that in regions where the kinematic vorticity number las 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 (-Q(S), R-S) 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.