Lagrangian triangle and tetrad statistics in isotropic turbulence

We study the displacement statistics of three- and four-particle clusters extracted from direct numerical simulations of three-dimensional isotropic turbulence at Reynolds numbers ranging from \(R_{\lambda} \approx 240\) to 650. These statistics determine the third and fourth moments, respectively, of scalar concentration fields. Our focus is on the nature of non-Gaussian dynamics expressed via the shape factors \(I_i=g_i/R^2,(i=1,2,3)\) which are defined in terms of the eigenvalues \(g_i\) of the moment-of-inertia tensor and the radius of gyration \(R\), which represents the linear size of the cluster. Shape factors computed from clusters with initial sizes in the inertial sub-range approach constant values at intermediate times. The average values obtained, \(\langle I_1 \rangle \approx 0.83\), \(\langle I_2 \rangle \approx 0.16\) and \(\langle I_3 \rangle \approx 0.015\) for four-particle clusters, are insensitive to Reynolds number in the present data range, possibly indicating an approach to self-similar inertial sub-range behavior. These results differ from their respective Gaussian values of 0.75, 0.22 and 0.03. High-order statistics conditioned on cluster size are used to explore the nature and origins of these departures from Gaussian behavior and guide development of maximum-entropy theories of cluster shape.