Scalar flux

the closure problem reduces to finding an appropriate expression for the scalar flux (Uj(p). In high-Reynolds-number turbulent flows, the molecular transport term is again negligible. Thus, the scalar-flux term is responsible for the rapid mixing observed in turbulent flows. 

Like the Reynolds stresses, the scalar flux obeys a transport equation that can be derived from the Navier-Stokes and scalar transport equations. We will first derive the transport equation for the scalar flux of an inert scalar from (2.99), p. 48, and the governing equation for inert-scalar fluctuations. The latter is found by subtracting (3.89) from (1.28) (p. 16), and is given by 

The derivation of the scalar-flux transport equation proceeds in exactly the same manner as with the Reynolds stresses. We first multiply (2.99), p. 48, by (p and (3.90) by ut. By adding the resulting two expressions, we find 

This expression can be further simplified by making use of the following identities  

Reynolds averaging of (3.91) then yields the transport equation for the inert-scalar flux 27 

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Once again, these fluxes are not all independent and some care must be taken to rewrite everything so that syimnetry is preserved [12]. Wlien this is done, the Curie principle decouples the vectorial forces from the scalar fluxes and vice versa [9]. Nevertheless, the reaction temis lead to additional reciprocal relations because... 

Returning to Eq. (166), the third term on the left-hand side involves the turbulent scalar fluxes, defined by... 

Likewise, important joint velocity, composition statistics can be computed from the one-point joint velocity, scalar PDF. For example, the scalar flux is defined by... 

Thus, 4> (k, t) d roughly corresponds to the amount of scalar variance located at point k in wavenumber space at time t. Similar statements can be made concerning the relationship between and the scalar flux (ut(p), and between [Pg.90]

Similar relationships exist for the scalar flux energy spectrum and the scalar covariance energy spectrum. 

This expression applies to the case where there is no mean scalar gradient. Adding a uniform mean scalar gradient generates an additional source term on the right-hand side involving the scalar-flux energy spectrum. 

Furthermore, space and time derivatives of mean quantities can be easily related to space and time derivatives of /u>(V, t/e x, l). For example, starting from (3.84), the time derivative of the scalar flux is given by... 

The first term on the right-hand side of this expression is the molecular transport term that scales as Sc Re 1. Thus, at high Reynolds numbers,26 it can be neglected. The two new unclosed terms in (3.88) are the scalar flux (u.ja), and the mean chemical source term (Sa(chemical reacting flows, the modeling of (Sa(0)) is of greatest concern, and we discuss this aspect in detail in Chapter 5. 

The final term on the right-hand side of (3.95) is the scalar-flux dissipation sf defined by... 

In locally isotropic turbulence, the fluctuating velocity gradient and scalar gradient will be uncorrelated, and sf will be null. Thus, at sufficiently high Reynolds number, the scalar-flux dissipation is negligible. 

The transport equation for the scalar flux of a reacting scalar [Pg.103]

Thus, (3.105) has three unclosed terms the scalar flux Uj), the scalar variance flux (Uj(p/2), and the scalar dissipation rate e, defined by... 

Furthermore, in stationary isotropic turbulence the scalar flux is related to the mean scalar gradient by... 

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 last term on the right-hand side is unclosed and represents scalar transport due to velocity fluctuations. The turbulent scalar flux ( , varies on length scales on the order of the turbulence integral scales Lu, and hence is independent of molecular properties (i.e., v and T).17 In a CFD calculation, this implies that the grid size needed to resolve (4.70) must be proportional to the integral scale, and not the Batchelor scale as required in DNS. In this section, we look at two types of models for the scalar flux. The first is an extension of turbulent-viscosity-based models to describe the scalar field, while the second is a second-order model that is used in conjunction with Reynolds-stress models. 

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 scalar flux is then proportional to a turbulent-diffusion coefficient ... 

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... 

This type of model is usually referred to as an algebraic scalar-flux model. Similarmodels for the Reynolds-stress tensor are referred to as algebraic second-moment (ASM) closures. They can be derived from the scalar-flux transport equation by ignoring time-dependent and spatial-transport terms. 

Like the Reynolds stresses, the scalar flux obeys a transport equation that was derived in Section 3.3 ... 

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