Dissipation range scalar

However, DNS data for Schmidt numbers near unity suggest that (3.70) provides the best model for the scalar-dissipation range (Yeung et al. 2002). 

For a passive scalar, the turbulent flow will be unaffected by the presence of the scalar. This implies that for wavenumbers above the scalar dissipation range, the characteristic time scale for scalar spectral transport should be equal to that for velocity spectral transport tst defined by (2.67), p. 42. Thus, by equating the scalar and velocity spectral transport time scales, we have23 t)... 

The scalar-dissipation wavenumber /cd is defined in terms of /cdi by /cd = Sc1/2kdi-Like the fraction of the turbulent kinetic energy in the dissipation range kn ((2.139), p. 54), for a fully developed scalar spectrum the fraction of scalar variance in the scalar dissipation range scales with Reynolds number as... 

Thus, like the turbulence dissipation rate, the scalar dissipation rate of an inert scalar is primarily determined by the rate at which spectral energy enters the scalar dissipation range. Most engineering models for the scalar dissipation rate attempt to describe (kd, t) in terms of one-point turbulence statistics. We look at some of these models in Chapter 4. 

In principle, the same equation could be used for a reacting scalar. However, one would need to know the spectral distribution of the covariance chemical source term Sai, (3.141), in order to add the corresponding covariance-dissipation-range chemical source term to (3.165). 

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

The condition that a must be positive limits the applicability of the model to 1 < CMRei or 12 < R>,. This corresponds to k = ku = 0.1 k, so that scalar energy is transferred directly from the lowest-wavenumber band to the dissipative range. However, at such low Reynolds numbers, the spectral transfer rates used in the model cannot be expected to be accurate. In particular, the value of Rq would need to account for low-Reynolds-number effects. 

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 4.13. Predictions of the SR model for ReA = 90 and Sc = 1 for homogeneous scalar mixing with = 0. For these initial conditions, all scalar energy is in the dissipative range. Curves 1-3 and D correspond to the fraction of scalar energy in each wavenumber band

These variables are governed by exactly the same model equations (e.g., (4.103)) as the scalar variances (inter-scale transfer at scales larger than the dissipation scale thus conserves scalar correlation), except for the dissipation range (e.g., (4.106)), where... 

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. 

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,... 

The terms on the right-hand side of this expression are the unknown covariance source terms, in particular wavenumber bands due to chemical reactions. For example, for a fast non-premixed, one-step reaction one would expect the product to be formed in the scalar dissipation range, so that (Sa Q is dominant. 

Note that this model is exact if Gaa a Eaa in the scalar dissipation range. 

The second term on the right-hand side of (A.41) represents the flux of scalar dissipation into the scalar dissipation range, and can be rewritten in terms of known quantities. From (A.39), it can be seen that Taa(icD, t) = TD(t). Likewise, using the definition of kd, it follows that 2raK = CD(e/v)1/2. The scalar-dissipation flux term can thus be expressed as... 

The final term in (A.41) (Daa) is modeled by the product of the inverse of a characteristic time scale for the scalar dissipation range and eau- The characteristic time scale is taken to be proportional to ,2)i)/ u so that the final term has the form Cd 2D/(2)D. The proportionality constant Cd is thus defined by... 

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