**3. Subgrid‐scale scalar flux models**

In the context of mixing of scalars, the interaction between resolved and nonresolved scalar structures is accounted for by the unknown SGS scalar flux term, which must be provided via SGS scalar flux models. In the context of mixing, the aim of LES closures is to express the SGS scalar flux in terms of the known filtered values in order to close the numerical set or partial differential equations. Accounting for the nature of turbulent mixing, where various regimes are present in the scalar spectrum, modeling of the SGS scalar flux models is a complex task that is far from trivial [45].

Macromixing, mesomixing, and micromixing happen simultaneously as the mixing process is taking place. The spatial and temporal transport of large‐scale structures describes the macromixing, and they can be solved in LES. Mesomixing is driven by turbulent fluctua‐ tions in the energy‐containing range, and viscous convective deformations of fluid elements and molecular diffusion are responsible mechanisms for micromixing. Because the inertial range of the velocity spectrum is modeled in LES, mesomixing can be expressed by gradient correlations. On the other hand, characterization of micromixing can be based on the second statistical moment (variance) of the local concentration distribution [46–48]. A physical meaning of the concentration variance is that it provides a measure of the scalar distribution from small‐scale homogeneity. The variance production, then, is dependent on the scalar flux.

There are many published works proposing SGS scalar flux models; most of them were developed for gas flow. For review of SGS scalar flux models in gas flows, see Refs. [27, 49]. Even though gas‐based SGS scalar flux models have some limitations on the prediction of mixing where high Schmidt number effects are important [48], physically based models for

the SGS scalar flux have been rarely proposed. One exception is the study of Jaberi and Colucci [50].

According to experimental data and DNS results, the assumption of isotropy at inertial and dissipation scales of the scalar field is no longer valid for structured functions and derivative skewness, when a mean scalar gradient is taking place [4, 8, 35]. Different models have been proposed in order to take into account the anisotropic behavior of micromixing, mainly focused on gas‐phase (see Refs. [50–52]), but their application for fluids with high *Sc* num‐ bers is scarce.

Few models have been developed or used for high Schmidt flows. The eddy diffusivity model [53] with constant and dynamically calculated turbulent Schmidt number [39, 54] has been widely used on the simulation of mixing at high Schmidt numbers [40, 55].

Other models used in the simulation of high Schmidt flows have been adapted from SGS stress tensors or extended from their RANS counterpart. These models include the following:


On the other hand, the anisotropy model [48] was proposed for the simulation of mixing of passive and active scalars. Simulation results of mixing at low and high Schmidt regime have shown its superior performance among other SGS scalar flux models. The anisotropy and the eddy diffusivity models are presented in the following subsections.
