Spectral transport scalar

The model scalar energy spectrum was derived for the limiting case of a fully developed scalar spectrum. As mentioned at the end of Section 3.1, in many applications the scalar energy spectrum cannot be assumed to be in spectral equilibrium. This implies that the mechanical-to-scalar time-scale ratio will depend on how the scalar spectrum was initialized, i.e on E k, 0). In order to compute R for non-equilibrium scalar mixing, we can make use of models based on the scalar spectral transport equation described below. 

Like the turbulent energy spectrum discussed in Section 2.1, a transport equation can be derived for the scalar energy spectrum E (k, t) starting from (1.27) and (1.28) for an inert scalar (see McComb (1990) or Lesieur (1997) for details). The resulting equation is21 

the final term in (3.73) is responsible for scalar dissipation due to molecular diffusion at wavenumbers near kb.  

The scalar spectral transport equation can be easily extended to the cospectrum of two inert scalars Eap ic, t). The resulting equation is 

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

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The material covered in the appendices is provided as a supplement for readers interested in more detail than could be provided in the main text. Appendix A discusses the derivation of the spectral relaxation (SR) model starting from the scalar spectral transport equation. The SR model is introduced in Chapter 4 as a non-equilibrium model for the scalar dissipation rate. The material in Appendix A is an attempt to connect the model to a more fundamental description based on two-point spectral transport. This connection can be exploited to extract model parameters from direct-numerical simulation data of homogeneous turbulent scalar mixing (Fox and Yeung 1999). 

Because the integral scale is defined in terms of the energy spectrum, an appropriate starting point would be the scalar spectral transport equation given in Section 3.2. 

In order to understand better the physics of scalar spectral transport, it will again be useful to introduce the scalar spectral energy transfer rates T,f, and T p defined by... 

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)... 

Krt < k time scale defined in terms of the velocity spectrum (e.g., rst). 

Following the approach used to derive (2.75), p. 43, the scalar spectral transport equation can also be used to generate a spectral model for the scalar dissipation rate for the case 1 < Sc.24 Multiplying (3.73) by 2T/< 2 yields the spectral transport equation for D Ik, t) ... 

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