**3.1. The eddy diffusivity model**

(14)

(15)

(16)

The minimum corresponds to

406 Numerical Simulation - From Brain Imaging to Turbulent Flows

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

that is far from trivial [45].

The calculation of the parameter generates spatial and temporal fluctuations, as well as negative values [43]. Regions with a negative coefficient may be interpreted as regions where backscatter takes place. According to Carati et al. [44], the dynamic Smagorinsky model does not have information about the amount of energy that is available in the subgrid scales.

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

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

leading to

In analogy to the Smagorinsky model, most of the SGS scalar flux models are based on the eddy diffusivity assumption. The eddy diffusivity model [53] can be viewed as a counter‐ part of the Smagorinsky model for the scalar field. The eddy diffusivity model relates the SGS flux to the local filtered scalar strain rate, and the transport is assumed to be aligned to the filtered scalar gradient [15]. The proportionality constant is called the turbulent Schmidt number, which appears as an adjustable parameter. However, the turbulent Schmidt num‐ ber can be dynamically calculated by using the methodology proposed by Germano et al. [39] and further refined by Moin et al. [54].

$$J^{\nu \mu}\_{\alpha,j} = -C\_{\phi} \Delta^2 \left| \left\{ S \right\}\_L \right| \frac{\left\| \left< \phi \right>\_L}{\left< \alpha\_\gamma \right>} = -\frac{\nu\_\gamma}{\mathrm{Sc}\_\tau} \frac{\left< \left< \phi \right>\_L}{\left< \alpha\_\gamma \right>} \tag{17}$$

where *ScT* is the turbulent Schmidt number, which is given by the ratio of SGS viscosity to the SGS diffusivity (*ScT* =*vT* / *DT* ).

The eddy diffusivity model has been often used due to its simplicity, and it shows good results for gas flows. The model has been widely used in simple and complex geometries, some involving reactive scalars.

The turbulent Schmidt number appears as an adjustable parameter that can be tuned in order to minimize the error with reference data. According to Durbin and Patterson [63], the value is dependent on the type of flow, and is in the range 0.1–1 [27]. Various authors (see **Table 1**) have proposed different values for the nondimensional parameter *ScT* in high Schmidt flows.


**Table 1.** Turbulent Schmidt number used in LES of high Schmidt flows.

In turbulent shear flows, different orientations of the mean scalar gradient yield different values for the turbulent Schmidt number [49]. A constant value of *ScT* is, therefore, not adequate. The turbulent Schmidt number can be thought of as a parameter that characteriz‐ es the dissipative/diffusive cut‐off scales of the velocity and scalar fields. The behavior of the scalar mixing spectrum, presented in Section 2.2, indicates that a universal distribution of an effective turbulent Schmidt number cannot exist [27].

## **3.2. The eddy diffusivity model with dynamic procedure**

Similar to the dynamic Smagorinsky model, in the dynamic model, the *ScT* in Eq. (17) can be dynamically calculated by using the methodology proposed by Germano et al. [39], which was further extended to scalar transport and compressible flow by Moin et al. [54]. By applying a test filter over the filtered velocity and scalar fields, the turbulent Schmidt number can be computed as follows [64].

The filter scalar flux is

$$\frac{1}{\text{Sc}\_{T}} = \frac{1}{\text{2C}\_{\text{ohv}}} \frac{F\_{\prime}H\_{\prime}}{H\_{\prime}H\_{\prime}} \tag{18}$$

where

(17)

ber can be dynamically calculated by using the methodology proposed by Germano et al. [39]

where *ScT* is the turbulent Schmidt number, which is given by the ratio of SGS viscosity to the

The eddy diffusivity model has been often used due to its simplicity, and it shows good results for gas flows. The model has been widely used in simple and complex geometries, some

The turbulent Schmidt number appears as an adjustable parameter that can be tuned in order to minimize the error with reference data. According to Durbin and Patterson [63], the value is dependent on the type of flow, and is in the range 0.1–1 [27]. Various authors (see **Table 1**) have proposed different values for the nondimensional parameter *ScT* in high Schmidt flows.

In turbulent shear flows, different orientations of the mean scalar gradient yield different values for the turbulent Schmidt number [49]. A constant value of *ScT* is, therefore, not adequate. The turbulent Schmidt number can be thought of as a parameter that characteriz‐ es the dissipative/diffusive cut‐off scales of the velocity and scalar fields. The behavior of the scalar mixing spectrum, presented in Section 2.2, indicates that a universal distribution of an

0.8 [23] 0.7 [24]

0.4 [108]

[113], [114]

**Configuration** *ScT* **Reference**

Channel 0.25 [110] Inclined channel 0.7 [109]

Coaxial jet 0.7 [107] Jet in channel 1.0 [48] Mixing layer 1.0 [48] Co‐flowing jet 0.7 [99]

**Table 1.** Turbulent Schmidt number used in LES of high Schmidt flows.

effective turbulent Schmidt number cannot exist [27].

and further refined by Moin et al. [54].

408 Numerical Simulation - From Brain Imaging to Turbulent Flows

SGS diffusivity (*ScT* =*vT* / *DT* ).

involving reactive scalars.

Stirred tank reactors

Confined impinging jets

$$F\_i = \underline{Q}\_i - J\_i^{\text{sys}} = \left\langle u\_i \right\rangle\_L \left\langle \phi \right\rangle\_L - \left\langle u\_i \right\rangle\_L \left\langle \phi \right\rangle\_{L'} \quad H\_i = \Delta^2 \left| \left\langle S \right\rangle\_L \right| \frac{\left\langle \phi \right\rangle\_L}{\left\langle \chi\_i \right\rangle} - \Delta^2 \left| \left\langle S \right\rangle\_L \right| \frac{\left\langle \phi \right\rangle\_L}{\left\langle \chi\_i \right\rangle} \tag{19}$$


**Table 2.** Previous works of turbulent mixing of *Sc* >> 1 using the dynamic procedure.

On implementation of the dynamic procedure, negative values are clipped to zero, and a relaxation process is imposed to instantaneous values.

The dynamic procedure is one of the most popular SGS scalar flux models in LES of high Schmidt flows. **Table 2** summarizes the geometry and Schmidt numbers of previous works, using the dynamic eddy diffusivity model.

Theoretical considerations suggest that the dynamic procedure should be adequate for LES, where the scalar fluctuations are resolved but the velocity fluctuations are not, so that SGS kinetic energy transfer takes place [27]. This is the case of low Schmidt flows.

Since the eddy diffusivity model assumes the alignment of the scalar flux with the scalar gradient, this approach does not predict realistic values of the scalar flux components [45, 48]. In addition, errors are introduced, as the model does not account for the different dynamics of the velocity and scalar fields.

#### **3.3. The dynamic anisotropy model**

Another recent SGS scalar flux model is the anisotropy model [48]. Contrary to the eddy diffusivity and dynamic models, the anisotropic model [48] accounts for the nonlinear contributions of the SGS to the turbulent scalar flux. If it is assumed that the turbulent flow field (mean velocities and turbulence characteristics) is available from LES using the dynam‐ ic‐based Smargorinsky SGS model [39], the anisotropy model is the simplest explicit aniso‐ tropic‐resolving algebraic form, and consists of a cubic formulation in terms of the scalar gradients. This model is thermodynamically consistent as it agrees with the irreversibility requirement of the second law of thermodynamics [47, 65]. It combines the linear eddy diffusivity model with an additional term coupling the (deviatoric) SGS stress tensor and the gradient of the filtered scalar field. Pantangi et al. [65] present a detailed analysis:

$$J^{\text{sys}}\_{\alpha,i} = -D\_{\text{T}} \frac{\partial \{\phi\}\_{\text{L}}}{\partial \mathbf{x}\_{i}} + D\_{\text{dev}} T\_{\text{sys}} \pi^{\text{sys}(\text{dev})}\_{ij} \frac{\partial \{\phi\}\_{\text{L}}}{\partial \mathbf{x}\_{j}} \tag{21}$$

where *Ddev* is the anisotropy model coefficient. Both *DT* and *Ddev* depend on the invariants of *τij SGS* and <sup>∂</sup> *<sup>ϕ</sup> <sup>L</sup>* <sup>∂</sup> *xj* . The tensor diffusivity is defined by

$$D\_{i,j}^{\circ \circ \circ} = -D\_T \delta\_{\bar{\psi}} + D\_{akr} T\_{\psi \bar{\psi}} \pi\_{\bar{\psi}}^{\circ \circ (d \circ \circ)} \tag{22}$$

The anisotropy model may lead to other models proposed in the literature, according to the modeling level used for the deviatoric part of the SGS stress tensor. Note that the additional term is a measure of how local mixing, dependent on the molecular Schmidt number through the SGS timescale, is influenced by the nonresolved flow structures (see Eq. (9) in Ref. [65]).

If the turbulent flow field (mean velocities and turbulence characteristics) is available from the LES solver using the dynamic‐based Smargorinsky SGS model, the anisotropy model is the simplest explicit algebraic model that solved accounts for the anisotropy found in turbulent flows. In the context defined by the Smagorinsky model, the anisotropy model can be developed by expressing the SGS timescale using the filter size and the subgrid viscosity. The anisotropy model reads:

Transport and Mixing in Liquid Phase Using Large Eddy Simulation: A Review http://dx.doi.org/10.5772/63993 411

$$J^{\text{cyl}}\_{\alpha,i} = -\frac{\nu\_r}{\text{Sc}\_{\text{T}}} \frac{\partial \{\not\phi\}\_L}{\text{\"ax}\_i} + D\_{\text{ar}} \Delta^2 \overline{S^{\text{cyl}}\_{\text{\%}}} \frac{\partial \{\not\phi\}\_L}{\partial \mathbf{x}\_i} \tag{23}$$

The anisotropy coefficient, *Dan*, can be either specified or calculated by the dynamic proce‐ dure as well as the *ScT*. A preliminary evaluation of this procedure was reported in Ref. [47] for chemical liquid flows and for gases in Ref. [46]. In order to compute the anisotropy coefficient, the turbulent Schmidt number must be calculated using Eqs. (21)–(23). Then, the calculation of the anisotropy model coefficient is done as follows:

$$D\_{av} = \left(F\_i - \frac{C\_s}{\text{Sc}\_\gamma} H\_i\right) H\_{\gamma i} \; ; \; H\_{2i} = \Delta^2 \left| \left\{ S \right\}\_L \right| \frac{\left\| \left\{ \phi \right\} \_L}{\text{Ox}\_i} - \Delta^2 \left\{ S \right\}\_L \frac{\left\| \left\{ \phi \right\} \_L}{\text{Ox}\_i} \right\} \tag{24}$$
