Date Published: May 20, 2019
Publisher: Public Library of Science
Author(s): Nadeem A. Malik, Roi Gurka.
A theory for turbulent particle pair diffusion in the inertial subrange [Malik NA, PLoS ONE 13(10):e0202940 (2018)] is investigated numerically using a Lagrangian diffusion model, Kinematic Simulations [Kraichnan RH, Phys. Fluids 13:22 (1970); Malik NA, PLoS ONE 12(12):e0189917 (2017)]. All predictions of the theory are observed in flow fields with generalised energy spectra of the type, E(k) ∼ k−p. Most importantly, two non-Richardson regimes are observed: for short inertial subrange of size 102 the simulations yield quasi-local regimes for the pair diffusion coefficient, K(l)∼σl(1+p)/2; and for asymptotically infinite inertial subrange the simulations yield non-local regimes K(l)∼σlγ, with γ intermediate between the purely local scaling γl = (1 + p)/2 and the purely non-local scaling γnl = 2. For intermittent turbulence spectra, E(k) ∼ k−1.72, the simulations yield K∼σl1.556, in agreement with the revised 1926 dataset K∼σl1.564 [Richardson LF, Proc. Roy. Soc. Lond. A 100:709 (1926); Malik NA, PLoS ONE 13(10):e0202940 (2018)]. These results lend support to the physical picture proposed in the new theory that turbulent diffusion in the inertial subrange is governed by both local and non-local diffusion transport processes.
Turbulent transport and mixing play an essential role in many natural and industrial processes [1–9], where concentration fluctuations, which is related to the pair separation, often play a critical role. Most theories of turbulent particle pair diffusion assume Richardson’s locality hypothesis [10, 11]. However, a new theory for turbulent particle pair diffusion based on the physical picture that both local and non-local diffusional processes govern the pair diffusion process has been proposed by the author in .
In order to characterize the pair diffusion process, Richardson assumed a scale dependent pair diffusion coefficient (turbulent diffusivity), because convective gusts of winds increase the pair separation at different rates depending on the separation [13–15]. In 1926, from observational data of turbulent pair diffusion coefficients collected from different sources, he assumed an approximate constant power law fit to the data, K(l) ∼ l4/3, . This is equivalent to 〈l2〉 ∼ t3 [11, 16], and is often referred to as the Richardson-Obukov t3-regime. l(t) is the pair separation at time t and the angled brackets is the ensemble average over particle pairs. Note that the assumed 4/3-scaling is consistent with Kolmogorov turbulence K41 theory, see .
In  it was hypothesised that the non-local process could also be isolated and exposed by taking a very small initial separation l0 ≪ η. Then the early motion should be purely strain dominated relative motion so long as σl(t) ≪ η. It was also noted that this regime should be independent of the form of E(k) and also of the size of the inertial subrange. To examine this hypothesis we therefore only need to test a few cases to prove generality.
The numerical results presented here are the most comprehensive obtained to date from KS due to the very large ensemble of particle pairs and the small time steps used. The statistical fluctuations in the results are therefore small. The γ(p)’s, which are the slopes of the plots in Fig 5, can be determined to within 1% error. An exception is close to the singular limit p = 1 where the numerical errors can be large. An accurate estimate of this error can be obtained as follows, noting first that the error level in γ(p) is identical to the error level in Mγ.
Richardson conceived his scaling law to be applicable to real turbulence, not just a mathematical curiosity. The new theory, developed in  and tested numerically here, generalises Richardson’s scaling arguments and is also constructed to be applicable to real turbulence. The fundamentally new concept here is that turbulent pair diffusion is the convolution of local and non-local diffusional processes; and from this idea two limiting cases of non-Richardson pair diffusion regimes has been obtained.