Eddy diffusivity operator in homogeneous isotropic turbulence

We use the recently developed macroscopic forcing method [A. Mani and D. Park, Phys. Rev. Fluids 6, 054607 (2021).] to compute the scale-dependent eddy diffusivity characterizing ensemble-averaged scalar and momentum transport in incompressible homogeneous isotropic turbulence. For scales larger than the energy containing eddies, eddy diffusivity is found to be constant and consistent with the Boussinesq approximation. However, for small scales eddy diffusivity is found to vanish inversely proportional to the wave number. Behavior at all scales is reasonably captured by a nonlocal eddy diffusivity operator modeled as $D/\sqrt{\mathcal{I}-l^2{\nabla }^2}$, where $D$ is the eddy diffusivity in the Boussinesq limit, and $l$ is a constant on the order of the large-eddy length. These results present a direct measurement of eddy diffusivity in turbulence with implications in turbulence modeling.

Figure 1

(a) A snapshot of the HIT simulation representing a component of the velocity field. (b) The real part of macroscopic force $s$ for mode $k=4$. (c) A 2D snapshot of the instantaneous scalar field $c$ subject to the given flow and $s$. (d) The mean scalar field $\overline{c}$

Figure 2

Measured macroscopic closure operators vs wave number for transport of (a) scalars and (b) momentum as well as the corresponding eddy diffusivity operators as shown in (c) and (d).

Figure 3

Schematic of the self-similar turbulent round jet and its mean velocity profile. Symbols show the laser Doppler anemometry (LDA) data of Hussein et al. [13], the blue line shows the RANS prediction using the Prandtl mixing length model (PMLM), and the red line shows the RANS prediction using a closure operator of the form shown in Eq. (9) with coefficients scaled consistently with the PMLM prescription.