**4. The filtered density function (FDF) approach**

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

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

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:

<sup>∂</sup> *xj* . The tensor diffusivity is defined by

where *Ddev* is the anisotropy model coefficient. Both *DT* and *Ddev* depend on the invariants of

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

(21)

(22)

kinetic energy transfer takes place [27]. This is the case of low Schmidt flows.

of the velocity and scalar fields.

*τij*

*SGS* and <sup>∂</sup> *<sup>ϕ</sup> <sup>L</sup>*

anisotropy model reads:

**3.3. The dynamic anisotropy model**

410 Numerical Simulation - From Brain Imaging to Turbulent Flows

A second approach in the LES mixing context is to solve the joint probability function of the SGS scalars [66], named filtered density function (FDF), in conjunction with the filtered momentum equations from LES. In this approach a filtered density function (FDF) is used to quantify the probability to find a filtered variable of the flow (*Y*) in the range (*Y*\*–Δ*Y*/2, *Y*\* +Δ*Y*/2). The fundamental property of the FDF method is to account, in a probabilistic way, for the effects of SGS fluctuations. The advantage of the FDF approach over other methods is that the chemical source term appears in a closed form, so that the turbulent/chemistry interac‐ tion can be correctly included. The counterpart of the FDF equation in the RANS context is named probability density function (PDF) equation. The main difference between PDF and FDF is that temporal fluctuations over different flow realizations are characterized by the use of PDF, while the instantaneous subgrid‐scale fluctuations are characterized by the use of FDF [67]. The PDF is the expected value of the FDF in the limit of vanishing filter width [68].

The statistical information of the reactive scalar fields can be obtained explicitly from a transport equation. Pope [66] introduced the mathematical definition of the FDF transport equation. Pope [66] highlighted one of the major advantages of the FDF approach: the chemical reaction term in the FDF transport equation is closed; so, the modeling of this term is no longer required. Drozda et al. [69] present an overview of the state of progress in FDF.

The mass‐filtered mass density function (FDF) is defined as

$$F\_{\perp}(\mathbf{\varPsi};\mathbf{x},t) = \coprod\_{\mathbf{x}}^{\ast \alpha} G(\mathbf{x}-\mathbf{x}',t)\,\rho(\mathbf{x}',t)\, f'(\mathbf{\varPsi};\mathbf{x}',t)\mathbf{dx}'\tag{25}$$

where Ψ is the sample space variable for each composition and *f* ' is the fine‐grained joint FDF of compositions [16], defined as

$$f^{\prime\prime}(\boldsymbol{\Psi}^{\prime};\boldsymbol{x},t) \equiv \prod\_{a=1}^{n\_{\phi}} \mathcal{S}\left(\phi\_{\phi}\left[\boldsymbol{x},t\right] - \boldsymbol{\nu}\_{\phi}\right) = \mathcal{S}\left(\boldsymbol{\Phi}\left[\boldsymbol{x},t\right] - \boldsymbol{\Psi}^{\prime}\right) \tag{26}$$

Based on the properties of Dirac delta functions and, after some manipulation, the FDF equation is [70]

$$\frac{\partial F\_{\boldsymbol{L}}}{\partial t} + \frac{\partial}{\partial \boldsymbol{x}\_{\boldsymbol{j}}} \left[ \boldsymbol{F}\_{\boldsymbol{L}} \left\langle \boldsymbol{\mu}\_{\boldsymbol{j}} \middle| \boldsymbol{\mu} \right\rangle\_{\boldsymbol{L}} \right] = -\frac{\partial}{\partial \boldsymbol{\mathcal{W}}\_{\boldsymbol{\alpha}}} \left\langle \boldsymbol{F}\_{\boldsymbol{L}} \left\langle \frac{\mathbf{D} \boldsymbol{\phi}\_{\boldsymbol{\alpha}}}{\mathbf{D}t} \middle| \boldsymbol{\mathbf{\varmathcal{W}}} \right\rangle\_{\boldsymbol{L}} \right\rangle \tag{27}$$

The introduction of the scalar balance equation into Eq. (19) gives

$$\frac{\partial F\_L}{\partial t} + \frac{\partial}{\partial \mathbf{x}\_j} \left[ F\_L \left\{ u\_j \middle| \mathbf{w} \right\}\_L \right] + \frac{\partial \left[ F\_L S\_a \right]}{\partial \mathbf{w}\_a} = -\frac{\partial}{\partial \mathbf{w}\_a} \left| F\_L \left\langle -\frac{1}{\rho} \frac{\partial J\_a}{\partial \mathbf{x}\_j} \middle| \mathbf{w} \right\rangle\_L \right| \tag{28}$$

The zeroth, first, and second order moments of FDF are

.

$$\begin{aligned} \int F\_L \left( \Psi; \mathbf{x}, t \right) d\mathfrak{P} &= \left\langle \rho \right\rangle\_l \\ \int \left\langle \nu\_\alpha F\_L \left( \Psi; \mathbf{x}, t \right) d\mathfrak{P} &= \left\langle \rho \phi\_\alpha \right\rangle\_l = \left\langle \rho \right\rangle\_l \left\langle \phi\_\alpha \right\rangle\_l \\ \int \left\langle \nu\_\alpha \nu\_\beta F\_L \left( \Psi; \mathbf{x}, t \right) d\mathfrak{P} &= \left\langle \rho \right\rangle\_l \left\langle \phi\_\alpha \phi\_\beta \right\rangle\_L \end{aligned} \tag{29}$$

Conditionally filtered terms of equations are unclosed. The conditional advection term can be modeled by the conventional gradient diffusion [70]:

$$\left( \left\langle u\_{\boldsymbol{\cdot}} \middle| \boldsymbol{\nu} \right\rangle\_{\boldsymbol{\cdot}} - \left\langle u\_{\boldsymbol{\cdot}} \right\rangle\_{\boldsymbol{\cdot}} \right) F\_{\boldsymbol{L}} = -\left\langle \boldsymbol{\rho} \right\rangle\_{\boldsymbol{\cdot}} D\_{\boldsymbol{\cdot}} \frac{\partial}{\partial \mathbf{x}\_{\boldsymbol{\cdot}}} \bigg( \frac{F\_{\boldsymbol{L}}}{\left\langle \boldsymbol{\rho} \right\rangle\_{\boldsymbol{\cdot}}} \right) \tag{30}$$

where *DT* is the SGS diffusion coefficient. The conditional diffusion term in Eq. (30) accounts for transport in the physical space and mixing in the composition space. It has to be mod‐ eled by mixing models. The interaction by exchange with the mean model (IEM) or linear mean square estimation (LMSE) [71, 72] is described by Eq. (32) [70]:

$$\frac{\partial}{\partial\boldsymbol{\nu}\_{a}}\bigg[\left\langle-\frac{1}{\rho}\frac{\partial\boldsymbol{J}\_{a}}{\partial\boldsymbol{\alpha}\_{\boldsymbol{\prime}}}\bigg|\boldsymbol{\nu}\right\rangle\_{\boldsymbol{\nu}}\bigg]\boldsymbol{F}\_{\boldsymbol{L}}\bigg]=\frac{\partial}{\partial\boldsymbol{\alpha}\_{\boldsymbol{\prime}}}\bigg[\left\langle\rho\right\rangle\_{\boldsymbol{\ell}}\left\langle\boldsymbol{D}\right\rangle\_{\boldsymbol{\ell}}\frac{\partial}{\partial\boldsymbol{\alpha}\_{\boldsymbol{\prime}}}\bigg[\frac{\boldsymbol{F}\_{\boldsymbol{L}}}{\{\rho\}\_{\boldsymbol{\ell}}}\bigg]\bigg]+\frac{\partial}{\partial\boldsymbol{\nu}\_{a}}\bigg[\boldsymbol{\Omega}\_{\boldsymbol{w}}\left(\boldsymbol{\nu}\_{a}-\left\langle\boldsymbol{\phi}\_{a}\right\rangle\_{\boldsymbol{\ell}}\right)\boldsymbol{F}\_{\boldsymbol{L}}\bigg]\tag{31}$$

where *CΩ* is the frequency of mixing within the subgrid, and it can be modeled as

Transport and Mixing in Liquid Phase Using Large Eddy Simulation: A Review http://dx.doi.org/10.5772/63993 413

$$
\Omega\_m = \frac{1}{\tau\_m} = C\_\Omega \left( \left\langle D \right\rangle\_L + D\_\tau \right) \Big/ \Delta\_L^2 \tag{32}
$$

where *CΩ* is the SGS mixing time constant. The solution of the FDF equation by convention‐ al difference schemes is intractable due to the high‐dimensionality of the equation: the computational cost rises exponentially as the number of scalars increases [73]. To overcome this problem, stochastic methods have been used for the solution of the equation, because the computational cost rises linearly with the number of scalars [74].

(26)

(27)

(28)

(29)

(30)

(31)

Based on the properties of Dirac delta functions and, after some manipulation, the FDF

Conditionally filtered terms of equations are unclosed. The conditional advection term can be

where *DT* is the SGS diffusion coefficient. The conditional diffusion term in Eq. (30) accounts for transport in the physical space and mixing in the composition space. It has to be mod‐ eled by mixing models. The interaction by exchange with the mean model (IEM) or linear mean

where *CΩ* is the frequency of mixing within the subgrid, and it can be modeled as

The introduction of the scalar balance equation into Eq. (19) gives

The zeroth, first, and second order moments of FDF are

412 Numerical Simulation - From Brain Imaging to Turbulent Flows

modeled by the conventional gradient diffusion [70]:

square estimation (LMSE) [71, 72] is described by Eq. (32) [70]:

equation is [70]

.

The FDF approach also has a modeling requirement, because the conditional convective flux and conditional diffusion terms in the FDF transport equation are unknown. By deducing an equation for the conditional convective flux based on the velocity‐scalar filtered density function formulation, Ref. [75] showed that a gradient transport hypothesis model performs well under many conditions. This model is usually implemented in FDF methods.

On the other hand, the conditional diffusion term accounts for transport of the FDF in the physical space and mixing in the composition space by the action of molecular diffusivity, and it has to be modeled by mixing models. From a theoretical point of view, McDermott and Pope [30] developed a set of desirable properties that an ideal FDF mixing model should fulfill. These properties differ from the corresponding PDF mixing models (see Ref. [16]). Most of the FDF mixing models have been adapted from their RANS counterpart.

A widely used mixing model is the interaction by exchange with the mean (IEM) model [71, 72]. The IEM model has a deterministic origin, initially formulated for PDF methods in RANS. The IEM states that the stochastic particle only interacts with itself, and the rate of change is proportional to the distance to the mean in the scalar space. Equivalently, the model relaxes the solution in the composition space toward the mean value; the relaxation is done over a subgrid‐scale mixing time. The major drawback of the IEM model is that it preserves the shape of the PDF, avoiding the relaxation toward a Gaussian distribution [76], being an unphysical situation. However, as pointed by [77], large‐scale mixing in inhomogeneous turbulent mixing problems also affects the shape of the distribution, and then limitation of the preservation of the distribution becomes less critical.

Numerical solution of the RANS/PDF equations has shown that the IEM model does not have good predictive capabilities (see Refs. [78, 79] for comparison of the accuracy of different PDF mixing models). On the contrary, the IEM model performs reasonably well in FDF/LES simulations. It has been used in most LES/FDF studies (see Refs. [70, 74, 78, 80–84] and others). There are few LES/FDF works in which other mixing models were used.

Mitarai et al. [78] and Olbricht et al. [74] implemented the modified Curl model—MCurl [85], which is a particle interaction model based on Curl's model [86]. In the Curl model, the PDF of the composition field is described by N stochastic particles. Mixing results from the movement of these particles in the composition space. According to the "simple" model, the mixing processes take place randomly between two stochastic particles, and the resulting composition of the new particle in the following time step is taken as the mean of the preced‐ ing particles. The Curl model produces a correct decay rate, but it relaxes the PDF to a bell‐ shaped curve instead of the Gaussian shape. This problem can be reduced by using improved models. In the modified Curl model of [85], the particle gradually mixes with the other one at a random mixing rate (for each pair of particles), instead of the originally instantaneous mixing assumption.

The Euclidean minimum spanning tree model (EMST) from Ref. [87] was evaluated by Mitarai et al. [78] and Shetty et al. [77]. The EMST is a sophisticated particle interaction model, where local mixing in composition space occurs between the selected group of particles, in contrast with the previous models (IEM, MCurl, etc.). Briefly, a Euclidean minimum spanning tree is constructed on the ensemble of particles which are selected based on a defined state variable. The Euclidean minimum spanning tree is formed by minimizing the length of the spanning tree. Mixing occurs between pairs of particles connected by a common branch of the tree. The EMST model has a superior performance over other PDF mixing models; however, it de‐ mands more computational time. In comparison to the IEM and MCurl, the implementation in the numerical solvers is more complicated.

Cleary et al. [88] implemented the multiple mapping conditioning (MMC) mixing model [89]. In the MMC model, a reference space is mapped to the physical space in order to attain local mixing in the composition space. This is done by choosing one or more reference variables (e.g., mixture fraction, sensible enthalpy, scalar dissipation, and others). Mixing occurs between particles that are close in both physical and reference spaces. The main difference of the MMC model with the EMST model is that local mixing in the former model is indirectly enforced in the composition space by assuring localness in the reference space, while it is directly enforced in composition space in the latter model [90]. Therefore, the principle of independence of scalars is assured in MMC. An advantage of the MMC model in FDF is that the averaged joint scalar distribution can be well predicted by using fewer particles in the computational domain than other models. However, the model has two unknown constants, and the reference space has to be solved in LES.

A recent PDF mixing model is the parameterized scalar profile (PSP), developed by Meyer and Jenny [91] and used by Shetty et al. [77] in LES/FDF. In this approach, it is assumed that one‐ dimensional effects dominate the dynamics of molecular diffusion at molecular interfaces. The PSP model is based on a parameterization of one‐dimensional scalar profile in high Rey‐ nolds number flows. These scalar profiles are composed of scaled sinusoidal shaped sections, which move with the fluid within a reference frame. There are three parameters used by the scalar profiles; so, they become as additional properties of the particles. The latter makes the model very expensive, because three additional equations must be solved.

A mixing model developed in the LES framework is the fractal IEM (FIEM) [77]. The FIEM is a modified version of the IEM in which it is considered that the behavior of the system at the small scales is repeated at the large scales. It is done by splitting the control volume into many smaller control volumes, each one having the same size, allowing strong mixing in the smaller control volumes and further weak mixing in the larger control volume. The model uses two additional empirical parameters that account for the relative importance of the two mixing steps into the global mixing process. These constants are tuned by trial and error.

composition of the new particle in the following time step is taken as the mean of the preced‐ ing particles. The Curl model produces a correct decay rate, but it relaxes the PDF to a bell‐ shaped curve instead of the Gaussian shape. This problem can be reduced by using improved models. In the modified Curl model of [85], the particle gradually mixes with the other one at a random mixing rate (for each pair of particles), instead of the originally instantaneous mixing

The Euclidean minimum spanning tree model (EMST) from Ref. [87] was evaluated by Mitarai et al. [78] and Shetty et al. [77]. The EMST is a sophisticated particle interaction model, where local mixing in composition space occurs between the selected group of particles, in contrast with the previous models (IEM, MCurl, etc.). Briefly, a Euclidean minimum spanning tree is constructed on the ensemble of particles which are selected based on a defined state variable. The Euclidean minimum spanning tree is formed by minimizing the length of the spanning tree. Mixing occurs between pairs of particles connected by a common branch of the tree. The EMST model has a superior performance over other PDF mixing models; however, it de‐ mands more computational time. In comparison to the IEM and MCurl, the implementation

Cleary et al. [88] implemented the multiple mapping conditioning (MMC) mixing model [89]. In the MMC model, a reference space is mapped to the physical space in order to attain local mixing in the composition space. This is done by choosing one or more reference variables (e.g., mixture fraction, sensible enthalpy, scalar dissipation, and others). Mixing occurs between particles that are close in both physical and reference spaces. The main difference of the MMC model with the EMST model is that local mixing in the former model is indirectly enforced in the composition space by assuring localness in the reference space, while it is directly enforced in composition space in the latter model [90]. Therefore, the principle of independence of scalars is assured in MMC. An advantage of the MMC model in FDF is that the averaged joint scalar distribution can be well predicted by using fewer particles in the computational domain than other models. However, the model has two unknown constants,

A recent PDF mixing model is the parameterized scalar profile (PSP), developed by Meyer and Jenny [91] and used by Shetty et al. [77] in LES/FDF. In this approach, it is assumed that one‐ dimensional effects dominate the dynamics of molecular diffusion at molecular interfaces. The PSP model is based on a parameterization of one‐dimensional scalar profile in high Rey‐ nolds number flows. These scalar profiles are composed of scaled sinusoidal shaped sections, which move with the fluid within a reference frame. There are three parameters used by the scalar profiles; so, they become as additional properties of the particles. The latter makes the

A mixing model developed in the LES framework is the fractal IEM (FIEM) [77]. The FIEM is a modified version of the IEM in which it is considered that the behavior of the system at the small scales is repeated at the large scales. It is done by splitting the control volume into many smaller control volumes, each one having the same size, allowing strong mixing in the smaller control volumes and further weak mixing in the larger control volume. The model uses two

model very expensive, because three additional equations must be solved.

assumption.

in the numerical solvers is more complicated.

414 Numerical Simulation - From Brain Imaging to Turbulent Flows

and the reference space has to be solved in LES.

The performance of different mixing models was evaluated by Mitarai et al. [78]. These authors compared the numerical results of IEM, EMST, and MCurl in RANS/PDF and LES/FDF with DNS data of a gas diffusion flame with one‐step reaction. The predicted filtered temperature was in an acceptable agreement with DNS results; the results from MCurl and IEM models were similar, while a better agreement was achieved with the EMST model. The mean and variance of the mixture fraction were accurately predicted by all mixing models. Their results also showed that LES/FDF approach provides much better results than the RANS/PDF in which marked differences between PDF mixing models were observed.

Shetty et al. [77] compared the predictions with IEM, EMST, PSP, and the FIEM mixing models in LES/FDF when modeling a low Schmidt three‐stream turbulent jet. Comparisons with experimental data were done in the near‐field of the configuration [max. downstream location = 7.2 jet diameters]. The simulation results of the mean and RMS of the radial scalar distribution obtained with all models followed the experimental observations. However, the FIEM results were in better agreement with the mean experimental observations at the largest downstream location. Results showed fast mixing of the scalar of the inner jet and slow mixing of the scalar of the annular jet, but, conversely, this did not happen when the FIEM was used. Large scale motions play an important role in mixing at the unstable shear layers formed between the inner, annular, and co‐flowing streams, mainly due to entrainment processes. In the inner part of this jet, the scalar field may behave similar to the velocity field. Because the filtered velocity field is supposed to be the same for all tests [not showed in the chapter], little variations are expected in the performance of different mixing models, as was the case for the IEM, EMST, and PSP models. This suggests that the performance of the FIEM could be the result of artificial diffusion transport in the calculations.

Another aspect in the development of FDF is the fact that in LES there may be computation‐ al zones where the velocity field is locally fully resolved (the DNS limit). The mixing model has to take into account this aspect. McDermott and Pope [30] showed that the latter can be accomplished by using a mean drift term in the particle composition equation, rather than a random walk in physical space. The model developed by McDermott and Pope [30] correct‐ ly reduces to DNS in the limit of vanishing filter width and is able to deal with differential diffusion.

On a discrete representation of the FDF at a given time, the particles in the sample space correspond to particles in physical space, and distance between particles in the subgrid volume is small [67]. In a LES/FDF computational domain, the numerical solution of a FDF equation using a particle‐based method may involve 106–108 stochastic particles. Therefore, the scalar resolution increases and may approximate to that of DNS, depending on the Schmidt number.

According to Klimenko [92], in the DNS limit, when the physical distance between particles becomes infinitesimally small (particle‐based solutions of the FDF equation), many mixing models behave essentially the same. Furthermore, recalling the differences between PDF and FDF methods, the modeling requirements are different: in the RANS context, the PDF aims to

solve mean scalar values, and fluctuations about the local mean value must be represented by the mixing models; contrarily, spatially filtered scalar values are sought in the LES approach, and the role of mixing models is to deal with SGS fluctuations about the local spatially filtered scalar. For these reasons, it is expected that simple mixing models used by the FDF method can provide a good approximation [68, 93]. In contrast, RANS has higher levels of sensitivity on the mixing model selection; so, the closure of the conditional diffusive term is the main difficulty in the application of PDF methods [88].

For review of the FDF method, see Haworth [93] who presents a comprehensive paper about theoretical aspects, physical models, numerical algorithms, and applications of PDF‐FDF methods. Drozda et al. [69] present an overview of the state of progress of LES/FDF methods applied to the particular case of turbulent combustion.

The application of the FDF method to high Schmidt flows is scarce. Schwertfirm et al. [94–96] used the FDF approach to study high Schmidt number flows using the DNS of the flow field (a priori analysis). Schwertfirm et al. [94–96] showed for three different Schmidt numbers (*Sc* = 3, 25), that if a correct definition of the SGS mixing frequency is done then the statistical and instantaneous behavior of the scalar field can be well predicted by using the IEM mixing model. One important finding is that the SGS scalar variance transport can be neglected in the SGS mixing frequency definition.

Van Vliet et al. [97] applied the LES/FDF approach to a tubular reactor with a moderated Reynolds number (*Re* = 4000) and high Schmidt number (*Sc* = 2000). They used IEM mixing model. They compared the mean and variance concentration distribution in the axial direc‐ tion with a few experimental data, and without reaction (conserved scalar). From a qualita‐ tive point of view, simulation results are in agreement with experimental observations. However, the quantitative comparison of simulation results with experimental information shows that the simulated concentration (mean) decays faster. A similar behavior was exhibit‐ ed by the scalar variance, showing a poor agreement with experiments. The discrepancy between experiments and simulations indicates that the mixing rate is overpredicted due to an inaccurate estimation of the subgrid‐scale mixing time, rather than a feature of the IEM model. The performance of the reactor was studied by the LES/FDF approach by changing the Damköhler number over eight (8) orders of magnitude. A LES/FDF numerical study of a low‐ density polyethylene tubular reactor was done by Van Vliet et al. [81], at the vicinity of the initiator injection point. Multiple reactive scalars were solved (concentration and tempera‐ ture), and a mechanism of six chemical reactions was used.

Experimental studies indicate that in gas flows, the subgrid mixing time constant is not universal, having values between 0.6 and 3.1 in the RANS context [76]. Different values of the SGS mixing time constant have been reported, mostly in the range 2–10 for gas flows. For liquid flows, Van Vliet et al. [97] used a value of 3. Mejía et al. [98] evaluated four values (between 4 and 10) in a turbulent round jet. They found that as the subgrid mixing time constant increas‐ es, the jet decay rate as well as the mixing rate decreases. The nonuniversality of this con‐ stant can affect the predictive capability of LES.

The works of Van Vliet et al. [81, 97] and Schwertfirm et al. [94–96] demonstrate the feasibili‐ ty of performing (advanced) numerical simulations of systems of practical interest (e.g., industrial reactors), with minimum assumptions and complex chemistry. The numerical simulation is a valuable tool for designing, optimizing, and evaluating such processes. Their simulations showed that the calculation of the SGS mixing time, a common parameter in many mixing models, is an Achilles heel in their simulation of the high Schmidt flow, and it may be also the case for other high Schmidt systems.

solve mean scalar values, and fluctuations about the local mean value must be represented by the mixing models; contrarily, spatially filtered scalar values are sought in the LES approach, and the role of mixing models is to deal with SGS fluctuations about the local spatially filtered scalar. For these reasons, it is expected that simple mixing models used by the FDF method can provide a good approximation [68, 93]. In contrast, RANS has higher levels of sensitivity on the mixing model selection; so, the closure of the conditional diffusive term is the main

For review of the FDF method, see Haworth [93] who presents a comprehensive paper about theoretical aspects, physical models, numerical algorithms, and applications of PDF‐FDF methods. Drozda et al. [69] present an overview of the state of progress of LES/FDF methods

The application of the FDF method to high Schmidt flows is scarce. Schwertfirm et al. [94–96] used the FDF approach to study high Schmidt number flows using the DNS of the flow field (a priori analysis). Schwertfirm et al. [94–96] showed for three different Schmidt numbers (*Sc* = 3, 25), that if a correct definition of the SGS mixing frequency is done then the statistical and instantaneous behavior of the scalar field can be well predicted by using the IEM mixing model. One important finding is that the SGS scalar variance transport can be neglected in the SGS

Van Vliet et al. [97] applied the LES/FDF approach to a tubular reactor with a moderated Reynolds number (*Re* = 4000) and high Schmidt number (*Sc* = 2000). They used IEM mixing model. They compared the mean and variance concentration distribution in the axial direc‐ tion with a few experimental data, and without reaction (conserved scalar). From a qualita‐ tive point of view, simulation results are in agreement with experimental observations. However, the quantitative comparison of simulation results with experimental information shows that the simulated concentration (mean) decays faster. A similar behavior was exhibit‐ ed by the scalar variance, showing a poor agreement with experiments. The discrepancy between experiments and simulations indicates that the mixing rate is overpredicted due to an inaccurate estimation of the subgrid‐scale mixing time, rather than a feature of the IEM model. The performance of the reactor was studied by the LES/FDF approach by changing the Damköhler number over eight (8) orders of magnitude. A LES/FDF numerical study of a low‐ density polyethylene tubular reactor was done by Van Vliet et al. [81], at the vicinity of the initiator injection point. Multiple reactive scalars were solved (concentration and tempera‐

Experimental studies indicate that in gas flows, the subgrid mixing time constant is not universal, having values between 0.6 and 3.1 in the RANS context [76]. Different values of the SGS mixing time constant have been reported, mostly in the range 2–10 for gas flows. For liquid flows, Van Vliet et al. [97] used a value of 3. Mejía et al. [98] evaluated four values (between 4 and 10) in a turbulent round jet. They found that as the subgrid mixing time constant increas‐ es, the jet decay rate as well as the mixing rate decreases. The nonuniversality of this con‐

difficulty in the application of PDF methods [88].

416 Numerical Simulation - From Brain Imaging to Turbulent Flows

applied to the particular case of turbulent combustion.

ture), and a mechanism of six chemical reactions was used.

stant can affect the predictive capability of LES.

mixing frequency definition.
