Explicit expressions for eddy-diffusivity fields and effective large-scale advection in turbulent transport

Abstract

Large-scale transport is investigated in terms of new explicit expressions for eddy diffusivities and effective advection obtained from asymptotic perturbative methods. The carrier flow is formed by a large-scale component plus a small-scale contribution mimicking a turbulent flow. The scalar dynamics is observed in its pre-asymptotic regimes (i.e. on scales comparable to those of the large-scale velocity). The resulting eddy diffusivity is thus a tensor field which explicitly depends on the large-scale velocity. Small-scale interactions also cause the emergence of an effective large-scale (compressible) advection field which, as a result of the present study however, turns out to be of negligible importance. Two issues are addressed by means of Lagrangian simulations: quantifying the possible deterioration of the eddy-diffusivity/effective advection description by reducing to zero the spectral gap separating the large-scale velocity component from the small-scale component; comparing the accuracy of our closure against other simple, reasonable, options. Answering these questions is important in view of possible applications of our closure to tracer dispersion in environmental flows

Eddy diffusivities of inertial particles in random Gaussian flows

“…E(k 0 ) being the turbulent kinetic energy associated to the wave-number. In principle, the decay time T c would depend on k itself, tipically like 1/ k or 1/ k 2 [14][15][16]. However, since we are considering a single wavenumber flow, we can consider it as a constant.…”

Exact results on the large-scale stochastic transport of inertial particles including the Basset history term

“…In the case of pure exponential correlation, the amplitude of the flow in Eq. (D1) corresponds to an Orstein-Uhlenbeck process with variance equal to 4 and unity decay time 24 :…”

“…This technique indeed renders it possible to control the statistical properties of turbulence better than by other methods based on Direct Numerical Simulations (DNS) or chaotic flows. The advantage of having more controllability justifies why synthetic models are oftentimes utilized although some key differences from DNS are present, especially when it comes to particle transport [23][24][25][26] .…”

“…Our flow will be a stochastic one having well-defined statistical properties set a priori, in a similar manner to synthetic models for turbulence 23,24,26 . The value of the eddy diffusivity will thus be dependent on the very statistical properties of the shear flow, namely its time-space correlation function.…”

Eddy diffusivity of quasi-neutrally-buoyant inertial particles

“…In particular, the function P encodes the large-scale asymptotics of the full solution we are after. As usual in homogenization theory [20,21,29,46], we determine P by canceling secular terms from the expansion up to order O(ε 2 ). The upshot [30] is that P satisfies the diffusion equation…”