Dynamics of the perceived velocity gradient tensor and its modelling

We study the dynamics of the perceived velocity gradient tensor M constructed from four tracer particles that initially form a regular tetrad of size r0. The exact evolution equation of M, derived in our previous work [Yang et al., J. Fluid Mech. 897, A9 (2020)], contains several unclosed terms. Using numerical data, we compare the exact dynamics of M with the tetrad model [Chertkov et al., Phys. Fluids 11, 2394 (1999)]. In particular, we project the motion onto the (R, Q) plane, where R and Q are the third-and second-order invariants of M. When r0 is in the inertial range of scales of the turbulent cascade, we find that at very short times the tetrad model correctly describes the main features of the dynamics of M on the (R, Q) invariants plane. This suggests that at any instant of time, the unclosed pressure and viscous contributions to the Eulerian dynamics of M could be described by the nonlinear and the damping terms, respectively, in the tetrad model. However, after a time of order similar to 2 tau K, where tau K is the Kolmogorov timescale, the action of the unclosed pressure contribution to the dynamics changes very dramatically to become a strong damping term on the (R, Q) plane. This qualitative change in the dynamics occurs on a timescale that differs from T0, the turnover time at scale r0. In addition, the fluctuations around the mean are found to deviate from the short-correlated-white-noise assumption in the tetrad model, as the timescale of the fluctuation is closer to T0 than tau K.