**1. Introduction**

At molecular scale, the mixing of two or more fluids having different composition is driven by diffusion, a rather slow process. Turbulence increases the rate of mixing by the action of large‐scale motions, making higher the contact surface area between adjacent unmixed fluid "elements." Turbulent mixing plays an important role in many engineering, biological, and environmental applications. In the case of reactive systems, chemical reactions can only take place after reactants are mixed at molecular level; in many of such systems, the availability of

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mixed reactants at the molecular scale limits the reaction rate; consequently, the mixing process can be a controlling rate.

In the chemical and pharmaceutical industries, an inadequate mixing can raise the produc‐ tion costs due to the reduction in selectivity, low yield, undesired product accumulation, and scale‐up and process development problems. Some costs related to mixing are reported by Paul et al. [1]: in 1989, the cost of poor mixing in the U.S. chemical industry was estimated at \$1 to \$10 billion; yield losses of 5% due to poor mixing are typical. In the pharmaceutical industry, the costs due to lower yield and due to problems in scale‐up and process develop‐ ment are on the order of \$100 and \$500 million, respectively. In the 1980s, the pulp and paper industry reported savings averaging 10–15% by introducing medium consistency mixer technology in their processes.

Numerous studies have dealt with transport and mixing of passive scalars in systems where the Schmidt number (*Sc*) is close to one, as it is the case for gas flows in which the smallest scalar scale (the Obukhov‐Corrsin scale) is of the same order of magnitude or higher than the smallest flow scale, the Kolmogorov scale (see Refs. [2–5]). However, in many industrial and biological applications that take place in the liquid phase, the ratio of the fluid momentum diffusivity to molecular diffusivity of the scalar is much greater than 1; so, the scalar field holds finer structures than the velocity field (see Refs. [6–9]).The smallest scalar length‐scale when the Schmidt number is greater than 1 is named the Batchelor scale (*ηB*) and is much smaller than the Obukhov‐Corrsin scale, *ηoc*, and the smallest velocity length‐scale,—the Kolmogor‐ ov scale (*η*). The small dimension of the Batchelor scale makes accurate measurement of the instantaneous concentration in turbulent liquid flows scarce and only possible with rather sophisticated systems [10].

To overcome many of the difficulties found in mixture‐based processes, numerical simula‐ tion appears as an important tool for design and process improvement, as it can predict important information about the flow and scalar fields, which is hard to measure. When numerical simulations of industrial process are carried out, key requirements are the compu‐ tational efficiency, algorithm robustness, and accurate representation of the process.

Models that correctly represent the mixing processes in turbulent flows are important for the design of various engineering and environmental applications where turbulence plays an important role. While the current understanding of turbulence allows a reasonable descrip‐ tion of the flow field, a universal description of turbulent mixing processes remains a challenge [11, 12].

Direct numerical simulation (DNS) gives the most detailed information about the flow, as it resolves all the scalar length‐ and timescales by setting up the spatial resolution according to the smallest scalar/flow scale. No empirical closure or turbulence model assumptions are required. DNS has been useful for the study of transition and turbulent flows [13]. Because physics and chemistry are properly represented, DNS has also been used for the analysis of complex phenomena in combustion systems [14]. The use of DNS has been increasing as a complementary means of study of turbulent mixing process [15]. Sophisticated discretiza‐ tion schemes are used by DNS, imparting low flexibility when complex geometries are used [13]. The full‐scale resolution requirement makes very expensive the computational cost, and sometimes prohibitive, even for future computational capabilities. The computational cost rises as *Re*<sup>3</sup> . Approximately 99% of calculations are used to solve the dissipation scales [16]. For these reasons, DNS is not a viable choice for systems of practical importance at high Reynolds and Schmidt numbers.

mixed reactants at the molecular scale limits the reaction rate; consequently, the mixing process

In the chemical and pharmaceutical industries, an inadequate mixing can raise the produc‐ tion costs due to the reduction in selectivity, low yield, undesired product accumulation, and scale‐up and process development problems. Some costs related to mixing are reported by Paul et al. [1]: in 1989, the cost of poor mixing in the U.S. chemical industry was estimated at \$1 to \$10 billion; yield losses of 5% due to poor mixing are typical. In the pharmaceutical industry, the costs due to lower yield and due to problems in scale‐up and process develop‐ ment are on the order of \$100 and \$500 million, respectively. In the 1980s, the pulp and paper industry reported savings averaging 10–15% by introducing medium consistency mixer

Numerous studies have dealt with transport and mixing of passive scalars in systems where the Schmidt number (*Sc*) is close to one, as it is the case for gas flows in which the smallest scalar scale (the Obukhov‐Corrsin scale) is of the same order of magnitude or higher than the smallest flow scale, the Kolmogorov scale (see Refs. [2–5]). However, in many industrial and biological applications that take place in the liquid phase, the ratio of the fluid momentum diffusivity to molecular diffusivity of the scalar is much greater than 1; so, the scalar field holds finer structures than the velocity field (see Refs. [6–9]).The smallest scalar length‐scale when the Schmidt number is greater than 1 is named the Batchelor scale (*ηB*) and is much smaller than the Obukhov‐Corrsin scale, *ηoc*, and the smallest velocity length‐scale,—the Kolmogor‐ ov scale (*η*). The small dimension of the Batchelor scale makes accurate measurement of the instantaneous concentration in turbulent liquid flows scarce and only possible with rather

To overcome many of the difficulties found in mixture‐based processes, numerical simula‐ tion appears as an important tool for design and process improvement, as it can predict important information about the flow and scalar fields, which is hard to measure. When numerical simulations of industrial process are carried out, key requirements are the compu‐

Models that correctly represent the mixing processes in turbulent flows are important for the design of various engineering and environmental applications where turbulence plays an important role. While the current understanding of turbulence allows a reasonable descrip‐ tion of the flow field, a universal description of turbulent mixing processes remains a

Direct numerical simulation (DNS) gives the most detailed information about the flow, as it resolves all the scalar length‐ and timescales by setting up the spatial resolution according to the smallest scalar/flow scale. No empirical closure or turbulence model assumptions are required. DNS has been useful for the study of transition and turbulent flows [13]. Because physics and chemistry are properly represented, DNS has also been used for the analysis of complex phenomena in combustion systems [14]. The use of DNS has been increasing as a complementary means of study of turbulent mixing process [15]. Sophisticated discretiza‐ tion schemes are used by DNS, imparting low flexibility when complex geometries are used

tational efficiency, algorithm robustness, and accurate representation of the process.

can be a controlling rate.

400 Numerical Simulation - From Brain Imaging to Turbulent Flows

technology in their processes.

sophisticated systems [10].

challenge [11, 12].

A more realistic alternative to DNS is the utilization of spatial filtering or temporal ensemble averages, such as large eddy simulation (LES) and unsteady Reynolds averaged Navier‐Stokes equations (U‐RANS) [17]. When LES and U‐RANS formulations are used, additional terms appear in the transport equations. These terms need to be modeled. The information given by DNS has been useful to validate the unclosed terms of LES and RANS approaches [18].

RANS is based on the application of the Reynolds decomposition to any quantity *Q*(*x*, *t*)=*Q* ¯ (*x*, *t*) + *Q* '(*x*, *t*), where *Q* ¯ (*x*, *t*) is the mean of *Q*, and *Q* '(*x*, *t*) is the deviation from the mean. The temporal average must be done for time intervals greater than any flow time‐ scales. Averaged equations contain additional unknown terms: Reynolds stresses, scalar fluxes, and averaged source terms. These terms are unclosed, and turbulence models must be provided in order to close the mathematical system. In RANS, the largest scales are solved, and the turbulent spectrum information is provided by turbulence models. By using RANS, a good balance between results and computational cost is obtained. RANS is useful in engi‐ neering, especially for industrial and environmental sectors [19]. It is possible to perform parametric studies, considering its low computational cost. The main limitation of RANS models is that they only give limited information about turbulence, because all turbulent scales are modeled. Although RANS has achieved reasonable good precision for simple flows, in the reproduction of large‐scale organized and nonstationary turbulence structures, when strong streamline curvature or nongradient transport is present, RANS has achieved limited success [20]. Some inaccuracies arise from the turbulent viscosity hypothesis, the equation for *ε* and the nonuniversality of the model's constants which have to be tuned in order to improve the simulation results [16]. Although RANS provides a reasonable computational cost‐ precision ratio, it does not always predict the concentration fluctuation distributions [21] in turbulent reacting liquid flows; in the case of stirred tank reactors (STR), the turbulent kinetic energy is unpredicted in the impeller region and discharge stream [22–25].

On the other hand, LES can be viewed as an intermediate approach between DNS and RANS, because in the former all turbulent structures are calculated, while in the latter they are modeled. The idea of LES is to solve the large, nonuniversal, anisotropic turbulent scales and to model the small turbulent scales, which contain less kinetic energy and are nearly univer‐ sal and isotropic. These small scales are easier to model than the whole turbulent spectrum. The LES approach was proposed by Leonard [26]. A higher fidelity on the representation of the flow structure than RANS is expected, since the geometry‐dependent, large turbulent scales are calculated.

The cut‐off length must be proportional to the longitudinal integral length‐scale [16], *LEI*. Typically, 92% of the kinetic energy is resolved for a one‐dimensional flow and 80% for a three‐ dimensional flow [16]. The proportionality constant depends on the filter specification. The uncertainty about residual motions is lowered by reducing the filter width; therefore, constitutive LES equations are grid‐dependent, that is, LES is an incomplete model. Referenc‐ es [27–29] review the LES literature.

Compared to DNS, LES saves the computational cost of solving the lowerturbulent scales (99% of DNS calculations), and can be used for the simulation of systems of practical importance. However, it is more expensive than a *k‐ε* model for a given grid resolution.

In shearflows, LES prediction capabilities are diminished by the fact that in the viscous region, the energy‐containing structures are of the same order of magnitude as the viscous length‐ scales (*δ<sup>v</sup>* =*v ρ* / *τw*), and LES cannot solve them. To face this issue, it is possible to refine the computational mesh in near‐wall zones in order to consider the small coherent structures developed on those zones, but this refinement implies more computational effort. There are three additional possibilities: incorporating the fluid behavior in the wall region by wall models, simulating those zones with RANS (hybrid RANS‐LES), or incorporating wall‐ adapted SGS closures.

Most research efforts in LES for reactive flows are concentrated on the SGS fluxes and Favre‐ filtered source terms [30]. In diffusion flames, chemical reactions take place after mixing of reactive species is achieved at the smallest scales of turbulence (unresolved in LES); so, in LES the combustion process must be modeled.

According to Ref. [31], the uncertainty initially contained in the nonresolved scales is propa‐ gated to the resolved ones; hence, it is said that SGS modeling is not a well‐posed problem. LES results can be interpreted as a different realization of the flow. They could have the same statistical properties of the flow and may predict the same spatially organized structures but at a different location.

LES has been successfully applied to turbulent flows, in particular to complex geometries in liquid phase such as stirred tank reactors (see Refs. [10, 24, 32–34]). Because LES equations are unclosed, SGS models must be supplied in order to specify the SGS stresses, SGS scalar fluxes, and filtered reaction terms.

There are still unanswered questions about the behavior of high Schmidt turbulent flows. LES could be an alternative method for studying these flows, as long as the flow's physical behavior can be captured by the subgrid‐scale models [15]. The LES equations are presented in the following section.
