Scalar flux gradient-diffusion model

In order to proceed, we will need to close (3.174) by using a gradient-diffusion model for the scalar fluxes ... 

By analogy with the Smagorinsky model, the SGS scalar flux can be modeled using a gradient-diffusion model (Eidson 1985) ... 

The turbulent diffusivity defined by (4.74) is proportional to the turbulent viscosity defined by (4.46). Turbulent-diffusivity-based models for the scalar flux extend this idea to arbitrary mean scalar gradients. The standard gradient-diffusion model has the form... 

While inconsistent with the closure for the mixture-fraction PDF, (5.319) does yield the usual gradient-diffusion model for the scalar flux, i.e., for ( ,f). However, it will not predict the correct behavior in certain limiting cases, e.g., when the mixture-fraction mean is constant, but the mixture-fraction variance depends on x.128... 

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... 

A more sophisticated scalar-flux model could be employed in place of the gradient-diffusion model. However, given the degree of approximation inherent in the multi-environment model, it is probably unwarranted. 

As noted in Chapter 1, the composition PDF description utilizes the concept of turbulent diffusivity (Tt) to model the scalar flux. Thus, it corresponds to closure at the level of the k-e and gradient-diffusion models, and should be used with caution for flows that require closure at the level of the RSM and scalar-flux equation. In general, the velocity, composition PDF codes described in Section 7.4 should be used for flows that require second-order closures. On the other hand, Lagrangian composition codes are well suited for use with an LES description of turbulence. 

In the present version of the SR model, the fractions y, and yn are assumed to be time-independent functions of Rei and Sc. Likewise, the scalar-variance source term Va is closed with a gradient-diffusion model. The SR model could thus be further refined (with increased computational expense) by including an explicit model for the scalar-flux spectrum. 

This equation contains three new terms, namely flux of scalar variance, production of variance and dissipation of scalar variance, which require further modeling to close the equation. The flux terms are usually closed by invoking the gradient diffusion model (with turbulent Schmidt number, aj, of about 0.7). This modeled form is already incorporated in Eq. (5.21). The variance production term is modeled by invoking an analogy with turbulence energy production (Spalding, 1971) ... 

Closure of the mean scalar field equation requires a model for the scalar flux term. This term represents the scalar transport due to velocity fluctuations in the inertial subrange of the energy spectrum and is normally independent of the molecular diffusivity. The gradient diffusion model is often successfully employed (e.g., [15, 78, 2]) ... 

However, as discussed in chap 1.2.7, the gradient-diffusion models can fail because counter-gradient (or up>-gradient) transport may occur in certain occasions [15, 85], hence a full second-order closure for the scalar flux (1.468) can be a more accurate but costly alternative (e.g., [2, 78]). 

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 ... 

In order to understand the physical basis for turbulent-diffusivity-based models for the scalar flux, we first consider a homogeneous turbulent flow with zero mean velocity gradient18 and a uniform mean scalar gradient (Taylor 1921). In this flow, velocity fluctuations of characteristic size... 

The turbulent diffusivity, Dt, is assumed to be proportional to the turbulent viscosity. 5q is the turbulent Schmidt number with a typical value of 0.7. Equation (12.5.2-1) assumes that the scalar flux and the mean species concentration gradient are aligned, in contrast with the scalar flux model (Annex 12.5.2.A). This is strictly valid only for isotropic turbulence. Nevertheless, (12.5.2-1) is frequently applied in CFD codes. 

In Eq. (1), Ja is the diffusive flux of species A (mass or moles per unit area per unit time), a vector quantity Dab is the binary or mutual diffusion tensor describing the diffusion of A in a mixture of A and B and Vc a is the spatial concentration gradient of A. If diffusion is isotropic, then it may be characterized by a scalar value Z ah- Pick s law is a phenomenological description of diffusion on a macroscopic scale. It is useful in the design and analysis of processes like chromatography that involve nonequilibrium mass transport, as mathematical models of chromatography concern themselves with how fast a solute penetrates into the stationary phase, i.e., the flux. 

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