Statistical description of turbulent dispersion

Abstract

We derive a comprehensive statistical model for dispersion of passive or almost passive admixture particles such as fine particulate matter, aerosols, smoke, and fumes in turbulent flow. The model rests on the Markov limit for particle velocity. It is in accordance with the asymptotic structure of turbulence at large Reynolds number as described by Kolmogorov. The model consists of Langevin and diffusion equations in which the damping and diffusivity are expressed by expansions in powers of the reciprocal Kolmogorov constant C_{0}. We derive solutions of O(C_{0}^{0}) and O(C_{0}^{-1}). We truncate at O(C_{0}^{-2}) which is shown to result in an error of a few percentages in predicted dispersion statistics for representative cases of turbulent flow. We reveal analogies and remarkable differences between the solutions of classical statistical mechanics and those of statistical turbulence.

Lagrangian statistics of turbulent channel flow at Reτ = 950 calculated with direct numerical simulation and Langevin models

“…2,17 However, the validity of this model has long been uncertain, because it was one of the many possible formulations of the damping function which satisfy the well-mixed criterion. 2,6,9,17 The C −1 0 -expansion brought clarity to this issue, 9 since it showed that the statistics of particle displacement obtained through application of the model according to Eq. (7) is unique up to a relative error of C −2 0 , see Brouwers.…”

“…9 The statistical model according to (8) and (9) implies Gaussianity of (d/dt)v i (t). The effects of intermittency apparent in particle accelerations are thus not captured.…”

“…(7) with a truncation error of order C −1 0 in statistical properties of particle velocity, but of order C −2 0 in statistical properties of particle displacement. 9 During the short times of correlation, the marked particle will have hardly moved so that a locally homogeneous field can be assumed to exist. The model according to the Hamiltonian base case can be described as…”

“…They have been compared with the results of time simulations of fluctuation Eqs. (6), (8), and (9). Values for the Eulerian quantities u 0 1 (x), (x), and σ ij (x) in these equations have been calculated from the DNS database.…”

“…4,6 Recently, a way out of the non-uniqueness problem has been proposed, which is based on an expansion in the reciprocal of the universal Kolmogorov constant C 0 and invokes Onsager symmetry. [7][8][9] This expansion is valid if the Kolmogorov constant is sufficiently large, which requires a rather high Reynolds number. Moreover, the Langevin model disregards the effects of viscosity at small time scales, which are appreciable at limited values of the Reynolds number.…”