A new assessment of the second-order moment of Lagrangian velocity increments in turbulence

Abstract

The behavior of the second-order Lagrangian structure functions on state-of-the-art numerical data both in two and three dimensions is studied. On the basis of a phenomenological connection between Eulerian space-fluctuations and the Lagrangian time-fluctuations, it is possible to rephrase the Kolmogorov 4/5-law into a relation predicting the linear (in time) scaling for the second order Lagrangian structure function. When such a function is directly observed on current experimental or numerical data, it does not clearly display a scaling regime. A parameterization of the Lagrangian structure functions based on Batchelor model is introduced and tested on data for 3d turbulence, and for 2d turbulence in the inverse cascade regime. Such parameterization supports the idea, previously suggested, that both Eulerian and Lagrangian data are consistent with a linear scaling plus finite-Reynolds number effects affecting the small-and large-time scales. When largetime saturation effects are properly accounted for, compensated plots show a detectable plateau already at the available Reynolds number. Furthermore, this parameterization allows us to make quantitative predictions on the Reynolds number value for which Lagrangian structure functions are expected to display a scaling region. Finally, we show that this is also sufficient to predict the anomalous dependency of the normalized root mean squared acceleration as a function of the Reynolds number, without fitting parameters.2

Lagrangian statistics for Navier–Stokes turbulence under Fourier-mode reduction: fractal and homogeneous decimations

“…In section 2, we introduce the model equations for the Eulerian and Lagrangian dynamics, as well as the decimation protocols; we also define the set-up of the numerical experiments performed, together with the relevant parameters. In section 3 we separately discuss the main results for the velocity field in terms of the Eulerian and Lagrangian statistical properties; while in section 4 we combine them together by quantitatively assessing the validity of the bridge relation [16][17][18][19][20][21]. Summary and discussions are contained in the last section.…”

Phase and precession evolution in the Burgers equation

“…For the numerical simulation we used Adams-Bashforth schemes of fourth order, with a number of collocation points ranged between N = 2 16 and N = 2 19 , and a time step δt ∼ 10 −5 . See Table 1 for more details about the numerical data.…”

“…This is particularly important for the evolution of large-scale structures in the Universe [9,10,11,12,13]. Burgers equation was originally conceived as a toy model for turbulence and it is frequently used as a testing ground for numerical schemes and as a training ground for developing mathematical tools to study Navier-Stokes turbulence and other hydrodynamical or Lagrangian problems [14,15,16,17,18,19]. Burgers equation represents one of the simplest nonlinear partial differential equations known to display a nontrivial scaling of the velocity field correlation functions.…”

How tracer particles sample the complexity of turbulence

“…This may provide an explanation of a reduced scaling range of Lagrangian structure functions compared to their Eulerian counterparts; for instance, it was argued that even for the second-order (i.e. the lowest non-trivial order) Lagrangian structure function a clear scaling range is only expected starting at R 5 10 3 »ĺ [34] or even beyond R 3 10 4 »ĺ [35], where Eulerian scaling is already well established. The presented approach is very general and therefore opens avenues to a new generation of models for transport in complex particle-laden flows by allowing an assessment of particle statistics from the knowledge of the flow field statistics only.…”