Lagrangian multi-particle statistics

Abstract

Combined measurements of the Lagrangian evolution of particle constellations and the coarse grained velocity derivative tensor $\partial \tilde u_i /\partial x_j$ are presented. The data is obtained from three dimensional particle tracking measurements in a quasi isotropic turbulent flow at intermediate Reynolds number. Particle constellations are followed for as long as one integral time and for several Batchelor times. We suggest a method to obtain quantitatively accurate $\partial \tilde u_i /\partial x_j$ from velocity measurements at discrete points. We obtain good scaling with $t_* = \sqrt {2r^2 /15S_r (r)}$ for filtered strain and vorticity and present filtered R-Q invariant maps with the typical ‘tear drop’ shape that is known from velocity gradients at viscous scales. Lagrangian result are given for the growth of particle pairs, triangles and tetrahedra. We find that their principal axes are preferentially oriented with the eigenframe of coarse grained strain, just like constellations with infinitesimal separations are known to do. The compensated separation rate is found to be close to its viscous counterpart as $1/2{\text{ }}(dr^2 /dt)/r^2 \cdot t_* /\sqrt 2 \approx 0.11 - 0.14$. It appears that the contribution from the coarse grained strain field, $r_i r_j \tilde s_{ij}$ filtered at scale Δ = r, is responsible only for roughly 50% of the separation rate. The rest stems from contributions with scales Δ < r.