Turbulent transport, models scalars

A general overview of models for turbulent transport is presented in Chapter 4. The goal of this chapter is to familiarize the reader with the various closure models available in the literature. Because detailed treatments of this material are readily available in other texts (e.g., Pope 2000), the emphasis of Chapter 4 is on presenting the various models using notation that is consistent with the remainder of the book. However, despite its relative brevity, the importance of the material in Chapter 4 should not be underestimated. Indeed, all of the reacting-flow models presented in subsequent chapters depend on accurate predictions of the turbulent flow field. With this caveat in mind, readers conversant with turbulent transport models of non-reacting scalars may wish to proceed directly to Chapter 5. 

This chapter is devoted to methods for describing the turbulent transport of passive scalars. The basic transport equations resulting from Reynolds averaging have been derived in earlier chapters and contain unclosed terms that must be modeled. Thus the available models for these terms are the primary focus of this chapter. However, to begin the discussion, we first review transport models based on the direct numerical simulation of the Navier-Stokes equation, and other models that do not require one-point closures. The presentation of turbulent transport models in this chapter is not intended to be comprehensive. Instead, the emphasis is on the differences between particular classes of models, and how they relate to models for turbulent reacting flow. A more detailed discussion of turbulent-flow models can be found in Pope (2000). For practical advice on choosing appropriate models for particular flows, the reader may wish to consult Wilcox (1993). 

The influence of a wall on the turbulent transport of scalar (species or enthalpy) at the wall can also be modeled using the wall function approach, similar to that described earlier for modeling momentum transport at the wall. It must be noted that the thermal or mass transfer boundary layer will, in general, be of different thickness than the momentum boundary layer and may change from fluid to fluid. For example, the thermal boundary layer of a high Prandtl number fluid (e.g. oil) is much less than its momentum boundary layer. The wall functions for the enthalpy equations in the form of temperature T can be written as ... 

Similarly, turbulent scalar transport models based on (1.28) for the case where the chemical source term is null have been widely studied. Because (1.28) in the absence... 

For example, the RTD can be computed from the results of a turbulent-scalar transport model, but not vice versa. 

As discussed in Section 2.1, in high-Reynolds-number turbulent flows the scalar dissipation rate is equal to the rate of energy transfer through the inertial range of the turbulence energy spectrum. The usual modeling approach is thus to use a transport equation for the transfer rate instead of the detailed balance equation for the dissipation rate derived from (1.27). Nevertheless, in order to understand better the small-scale physical phenomena that determine e, we will derive its transport equation starting from (2.99). 

Owing to the complexity of multi-point descriptions, almost all scalar transport models for complex flows are based on one-point statistics. As shown in Section 2.1, one-point turbulence statistics are found by integrating over the velocity sample space. Likewise,... 

In general, liquid-phase reactions (Sc > 1) and fast chemistry are beyond the range of DNS. The treatment of inhomogeneous flows (e.g., a chemical reactor) adds further restrictions. Thus, although DNS is a valuable tool for studying fundamentals,4 it is not a useful tool for chemical-reactor modeling. Nonetheless, much can be learned about scalar transport in turbulent flows from DNS. For example, valuable information about the effect of molecular diffusion on the joint scalar PDF can be easily extracted from a DNS simulation and used to validate the micromixing closures needed in other scalar transport models. 

In summary, in the equilibrium-chemistry limit, the computational problem associated with turbulent reacting flows is greatly simplified by employing the presumed mixture-fraction PDF method. Indeed, because the chemical source term usually leads to a stiff system of ODEs (see (5.151)) that are solved off-line, the equilibrium-chemistry limit significantly reduces the computational load needed to solve a turbulent-reacting-flow problem. In a CFD code, a second-order transport model for inert scalars such as those discussed in Chapter 3 is utilized to find ( ) and and the equifibrium com-... 

In Section 3.3, the general transport equations for the means, (3.88), and covariances, (3.136), of 0 are derived. These equations contain a number of unclosed terms that must be modeled. For high-Reynolds-number flows, we have seen that simple models are available for the turbulent transport terms (e.g., the gradient-diffusion model for the scalar fluxes). Invoking these models,134 the transport equations become... 

Relative to velocity, composition PDF codes, the turbulence and scalar transport models have a limited range of applicability. This can be partially overcome by using an LES description of the turbulence. However, consistent closure at the level of second-order RANS models requires the use of a velocity, composition PDF code. 

For laminar flow, the characteristic time of the fluid phase Tf can be deflned as the ratio between a characteristic velocity Uf and a characteristic dimension L. For example, in the case of channel flows confined within two parallel plates, L can be taken equal to the distance between the plates, whereas Uf can be the friction velocity. Another common choice is to base this calculation on the viscous scale, by dividing the kinematic viscosity of the fluid phase by the friction velocity squared. For turbulent flow, Tf is usually assumed to be the Kolmogorov time scale in the fluid phase. The dusty-gas model can be applied only when the particle relaxation time tends to zero (i.e. Stp 1). Under these conditions, Eq. (5.105) yields fluid flow. This typically happens when particles are very small and/or the continuous phase is highly viscous and/or the disperse-to-primary-phase density ratio is very small. The dusty-gas model assumes that there is only one particle velocity field, which is identical to that of the fluid. With this approach, preferential accumulation and segregation effects are clearly not predicted since particles are transported as scalars in the continuous phase. If the system is very dilute (one-way coupling), the properties of the continuous phase (i.e. density and viscosity) are assumed to be equal to those of the fluid. If the solid-particle concentration starts to have an influence on the fluid phase (two-way coupling), a modified density and viscosity for the continuous phase are generally introduced in Eq. (4.92). 

Turbulent Transport of Species and Heat. Modeling of the Scalar Flux... 

Turbulent diffusivity based closure models for the scalar fluxes describing turbulent transport of species relate the scalar flux to the mean species concentration gradient according to Reynolds analogy between turbulent momentum and mass transport. The standard gradient-diffusion model can be written ... 

Note that when solving the CFD transport equations, the mean velocity and turbulence state variables can be found independently from the mixture-fraction state variables. Likewise, when validating the CFD model predictions, the velocity and turbulence predictions can be measured in separate experiments (e.g., using particle-image velocimetry [PIV]) from the scalar field (e.g., using planar laser-induced fluorescence [PLIF]). 

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