Abstract
The motion of tracer particles is kinematically simulated around three elementary flow patterns; a Burgers vortex, a shear-layer structure with coincident vortices and a node-saddle topology. These patterns are representative for their broader class of coherent structures in turbulence. Therefore, examining the dispersion in these elementary structures can improve the general understanding of turbulent dispersion at short time scales. The shear-layer structure and the node-saddle topology exhibit similar pair dispersion statistics compared to the actual turbulent flow for times up to $3{-}10\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$, where, $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$ is the Kolmogorov time scale. However, oscillations are observed for the pair dispersion in the Burgers vortex. Furthermore, all three structures exhibit Batchelor’s scaling. Richardson’s scaling was observed for initial particle pair separations $r_{0}\leqslant 4\unicode[STIX]{x1D702}$ for the shear-layer topology and the node-saddle topology and was related to the formation of the particle sheets. Moreover, the material line orientation statistics for the shear-layer and node-saddle topology are similar to the actual turbulent flow statistics, up to at least $4\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$. However, only the shear-layer structure can explain the non-perpendicular preferential alignment between the material lines and the direction of the most compressive strain, as observed in actual turbulence. This behaviour is due to shear-layer vorticity, which rotates the particle sheet generated by straining motions and causes the particles to spread in the direction of compressive strain also. The material line statistics in the Burgers vortex clearly differ, due to the presence of two compressive principal straining directions as opposed to two stretching directions in the shear-layer structure and the node-saddle topology. The tetrad dispersion statistics for the shear-layer structure qualitatively capture the behaviour of the shape parameters as observed in actual turbulence. In particular, it shows the initial development towards planar shapes followed by a return to more volumetric tetrads at approximately $10\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$, which is associated with the particles approaching the vortices inside the shear layer. However, a large deviation is observed in such behaviour in the node-saddle topology and the Burgers vortex. It is concluded that the results for the Burgers vortex deviated the most from actual turbulence and the node-saddle topology dispersion exhibits some similarities, but does not capture the geometrical features associated with material lines and tetrad dispersion. Finally, the dispersion around the shear-layer structure shows many quantitative (until 2–$4\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$) and qualitative (until $20\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$) similarities to the actual turbulence.
Highlights
IntroductionThe role of the carrier fluid in the transport of scalar quantities (e.g. particles and chemicals) is important in many natural and industrial processes
The role of the carrier fluid in the transport of scalar quantities is important in many natural and industrial processes
The dispersion statistics are investigated by kinematically simulating the tracer particles around different structures, namely the shear-layer structure (SLS), the node-saddle topology (NST) and the Burgers vortex (BV) as explained in §§ 2 and 3
Summary
The role of the carrier fluid in the transport of scalar quantities (e.g. particles and chemicals) is important in many natural and industrial processes. Richardson’s t3 scaling of the mean-square pair separation has been found to directly follow Batchelor’s t2 scaling (Sawford et al 2008; Bitane, Homann & Bec 2012) for small initial separations and at relatively small time scales Due to this small time scale, the onset of Richardson’s regime can be explored qualitatively by studying dispersion around these structures. Goto & Vassilicos (2004) linked the dynamics of the two particle statistics to the flow around critical points in two-dimensional turbulence They showed that the particle pairs stayed together in the proximity of elliptical zero acceleration points (vortex centres) and separated in the vicinity of hyperbolic zero-acceleration points (saddle points).
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