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 tu rbulence, researchers have utilized the joint PDFs of invariants of the vel ocity gradient tensor to study the geometry of small-scale motions of turbu lence. However, the joint PDFs only give an instantaneous static representa tion of the properties of fluid particles and dynamical Lagrangian informat ion cannot be extracted. In this paper, the Lagrangian evolution of the inv ariants of the velocity gradient tensor is studied using conditional mean t rajectories (CMT). These CMT are derived using the concept of the condition al mean time rate of change of invariants calculated from a numerical simul ation 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-rotati on tensor show that the mean evolution in the (Sigma, Q(W)) phase plane, wh ere 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 th e (Sigma, Q(W)) phase plane suggest that the initial reduction of Q(W) in r egions of high Q(W) is due to viscous diffusion and that vorticity contract ion only plays a secondary role subsequent to this initial decay. It is als o found that in regions of the flow with small values of Q(W), the local va lues of Q(W) do not begin to increase, even in the presence of self-stretch ing, until a certain self-stretching rate threshold is reached, i.e. when S igma 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 kinemat ic 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 ev olve towards having a positive local value of the intermediate principal ra te of strain.