Mean scalar gradients

With mean scalar gradients, the scalar covariances and joint dissipation rates attain steady-state values found by setting the right-hand sides of (3.179) and (3.180) equal to zero. This yields 

For perfectly aligned mean scalar gradients, cos(0aP) = 1. Note also that in order for (3.184) to hold, 0 CaaCpp C, where the equality holds when the Schmidt numbers are equal. This condition has important ramifications when developing models for 

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Figure 3.7. Two random scalar fields (x, t ) as a function of x = xi with fixed t = t. The scalar fields were extracted from DNS of isotropic turbulence (R>. = 140, (U) = 0) with collinear uniform mean scalar gradients. Dashed line Sc = 1/8 solid line Sc = 1. The corresponding velocity field is shown in Fig. 2.1. (Courtesy of P. K. Yeung.)... 

Gaussian PDFs are found for homogeneous inert scalar mixing in the presence of a uniform mean scalar gradient. However, for turbulent reacting flows, the composition PDF is usually far from Gaussian due to the non-linear effects of chemical reactions. 

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. 

For homogeneous scalar mixing, the mean scalar gradient can be either null or constant, i.e., of the form ... 

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

Applying this expression in (3.115), and using the continuity equation for the mean velocity, yields Sf = 0. Thus, in high-Reynolds-number flows,29 Sf will be negligible. The mean-scalar-gradient term Gf is defined by... 

The Reynolds-number dependence of differential-diffusion effects on gap is distinctly different than on pap, and can be best understood by looking at scalars in homogeneous, stationary turbulence with and without uniform mean scalar gradients. 

We will now use these equations to find expressions for the correlation coefficients for cases with and without mean scalar gradients. 

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 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... 

Figure 4.9. Sketch of CSTR representation of the SR model for 1 < Sc. Each wavenumber band is assumed to be well mixed in the sense that it can be represented by a single variable

Cs = Cb - Co, Cb = 1, and Cd = 3 (Fox 1995).36 Note that at spectral equilibrium, Vp = p, % = To = p( I - i/i)), and (with Sc = 1) R = Rq. The right-hand side of (4.117) then yields (4.114). Also, it is important to recall that unlike (4.94), which models the flux of scalar energy into the dissipation range, (4.117) is a true small-scale model for p. For this reason, integral-scale terms involving the mean scalar gradients and the mean shear rate do not appear in (4.117). Instead, these effects must be accounted for in the model for the spectral transfer rates. 

Figure 4.14. Predictions of the multi-variate SR model for Re, = 90 and Sc = (1, 1/8) with collinear mean scalar gradients and no backscatter (cb = 0). For these initial conditions, the scalars are uncorrelated pap(0) = gap(0) = 0. The correlation coefficient for the dissipation range, pD, is included for comparison with pap. 

The case of uniform mean scalar gradients was introduced in Section 3.4, where Gia (see (3.176)) denotes the ith component of the gradient of (< In this section, we will assume that the mean scalar gradients are collinear so that GiaGip = GiaGia = G,pG,p = G2. The scalar covariance production term then reduces to V p = 2rTG2. In the absence of differential diffusion, the two scalars will become perfectly correlated in all wavenumber bands, i.e.,

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Note that, unlike pap, pD is not bounded by unity. This is because the dissipation-range cut-off wavenumber, (4.101), used to define ( a p) d depends on Scap, which is different for each scalar pair (i.e., Scap / Scaa Scpp). Indeed, for the mean-scalar-gradient case,... 

Multi-environment presumed PDF models are generally not recommended for homogeneous flows with uniform mean gradients. Indeed, their proper formulation will require the existence of a mixture-fraction vector that, by definition, cannot generate a uniform mean scalar gradient in a homogeneous flow. 

Note, however, that in the presence of a mean scalar gradient the local isotropy condition is known to be incorrect (see Warhaft (2000) for a review of this topic). Although most molecular mixing models do not account for it, the third constraint can be modified to... 

In this limit, molecular-scale terms can be neglected. The relationship (6.87) is exact for statistically homogeneous scalars in the absence of uniform mean scalar gradients. This is the case considered in Fox (1999), and Fox (1994), and can occur even in the absence of turbulence. 

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