Lagrangian multi-particle statistics

Abstract

Combined measurements of the Lagrangian evolution of particle constellations and the coarse-grained velocity derivative tensor ∂ũ _i /∂ x _j are presented. The data are obtained from three-dimensional particle tracking measurements in a quasi isotropic turbulent flow at an 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 ∂ũ _i /∂ x _j from velocity measurements at discrete points. Based on an analytical result and on a sensitivity analysis, both presented here, we estimate the accuracy for filtered strain, ᵴ ², and enstrophy, ², at around 30%. The accuracy improves with higher tracer seeding density and with smaller filter scale Δ. We obtain good scaling with t* = √2r ²/15S ₂(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 results are given for the growth of particle pairs, triangles and tetrahedra. 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⟨ (dr ²/dt)/r ²⟩· t _*/√2≈ 0.12. It appears that the contribution from the coarse-grained strain field, r _i r _j ᵴ _ij filtered at scale Δ = r, is responsible for roughly 2/3 of the separation rate, while 1/3 stems from scales Δ < r.

Keywords:

• Multi-particle statistics
• Velocity gradient
• Lagrangian evolution
• Particle tracking

Acknowledgements

This work was financed through grants of the Danish Research Agency (grant nr: 274-05-0540) and of the Swiss Science Foundation (grant nr: PA0024-4111444/1).