Multi-scale geometric analysis of Lagrangian structures in isotropic turbulence

Abstract

We report the multi-scale geometric analysis of Lagrangian structures in forced isotropic turbulence and also with a frozen turbulent field. A particle backwardtracking method, which is stable and topology preserving, was applied to obtain the Lagrangian scalar field φ governed by the pure advection equation in the Eulerian form ∂ t φ + u · ∇φ = 0. The temporal evolution of Lagrangian structures was first obtained by extracting iso-surfaces of φ with resolution 1024 3 at different times, from t = 0 to t = T e , where T e is the eddy turnover time. The surface area growth rate of the Lagrangian structure was quantified and the formation of stretched and rolled-up structures was observed in straining regions and stretched vortex tubes, respectively. The multi-scale geometric analysis of Bermejo-Moreno & Pullin (J. Fluid Mech., vol. 603, 2008, p. 101) has been applied to the evolution of φ to extract structures at different length scales and to characterize their non-local geometry in a space of reduced geometrical parameters. In this multi-scale sense, we observe, for the evolving turbulent velocity field, an evolutionary breakdown of initially large-scale Lagrangian structures that first distort and then either themselves are broken down or stretched laterally into sheets. Moreover, after a finite time, this progression appears to be insensible to the form of the initially smooth Lagrangian field. In comparison with the statistical geometry of instantaneous passive scalar and enstrophy fields in turbulence obtained by Bermejo-Moreno & Pullin (2008) and Bermejo-Moreno et al. (J. Fluid Mech., vol. 620, 2009, p. 121), Lagrangian structures tend to exhibit more prevalent sheet-like shapes at intermediate and small scales. For the frozen flow, the Lagrangian field appears to be attracted onto a stream-surface field and it develops less complex multi-scale geometry than found for the turbulent velocity field. In the latter case, there appears to be a tendency for the Lagrangian field to move towards a vortex-surface field of the evolving turbulent flow but this is mitigated by cumulative viscous effects.

On Lagrangian and vortex-surface fields for flows with Taylor–Green and Kida–Pelz initial conditions

“…Finally, this appears to move towards a stationary stage, which might be controlled by O (B(t)), as shown in figure 12 and implied in (5.6). In contrast, Yang et al (2010) showed that a Lagrangian field, starting from an arbitrary initial condition with a moderate |λ ω0 | in stationary isotropic turbulence, appeared to exhibit a tendency to move towards a vortex-surface field of the evolving flow, but this is mitigated by cumulative viscous effects. In the TG flow, from an arbitrary initial Lagrangian field φ 0 = sin x sin y sin z, we also observed a similar decreasing trend for |λ ω | in figure 12.…”

“…Here, N p should be greater than the grid number N of the velocity field, i.e. the resolution of the Lagrangian field should be higher than the velocity field to capture some nearly singular Lagrangian structures in the evolution (Yang et al 2010). Then, particles are released and their trajectories calculated by solving (4.2) backwards in time.…”

“…At the same time, the recent development of tracking methodologies for finite-sized Lagrangian structures (e.g. Lindsay & Krasny 2001, Goto & Kida 2007and Yang, Pullin & Bermejo-Moreno 2010 offer effective approaches to numerical simulation of the evolution of vortex surfaces in a high-Reynolds-number flow or even an inviscid flow with the aid of substantial computational resources.…”

Evolution of vortex-surface fields in viscous Taylor–Green and Kida–Pelz flows

“…Using (2.18) and ∂ω(x, t)/∂τ = 0 at fixed t, and following a line of analysis applied in Yang, Pullin & Bermejo-Moreno (2010) and to a Lagrangian field, we can show, after some algebra, that…”

Geometric study of Lagrangian and Eulerian structures in turbulent channel flow

“…This issue was discussed by Green, Rowley & Haller (2007) who showed the evolution of single vortex-like structures into a packet of similar structures in turbulent channel flow by identifying 'Lagrangian coherent structures' (Haller 2001). Yang, Pullin & Bermejo-Moreno (2010) illustrated and quantified the evolutionary geometry in the breakdown of initially large-scale Lagrangian structures in isotropic turbulence using a multi-scale geometric analysis.…”