Spectral transfer rates

The spectral transfer rates are assumed to be local in wavenumber space 37 

In principle, the forward and backward transfer rates can be computed directly from DNS (see Appendix A). However, they are more easily computed by assuming idealized forms for the scalar energy spectrum (Fox 1995). In the general formulation (Fox 1999), they include both a forward cascade (a) and backscatter (0)  

The forward cascade parameters can be determined from the fully developed isotropic velocity spectrum (Fox 1995) 38 

The Schmidt-number dependence of 0m is a result of the definition of / o, and has been verified using DNS (Fox and Yeung 1999). 

The initial conditions for ( / /2) and e are determined by the initial scalar spectrum. Examples of SR model predictions with four different initial conditions are shown in 

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In a fully developed turbulent flow,22 the scalar spectral transfer rate in the inertial-convective sub-range is equal to the scalar dissipation rate, i.e., T k) = for /cei < < Kn. Likewise, when Sc 1, so that a viscous-convective sub-range exists, the scalar trans-... 

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. 

Because the parameters 0(30, phu- and Pm appearing in TD depend on the Schmidt number, the dissipation-range spectral transfer rates will be different for each covariance component. Vap is the covariance-production term defined by (3.137) on p. 90. 

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