Spectral relaxation model

As an illustrative example, we will consider the SR model equations for Sc 1 and Rc , 100. (A general derivation of the model is given in Appendix A.) For this range of 

Reynolds numbers, the inertial-convective sub-range contains two stages. The wavenumber bands are defined by the cut-off wavenumbers 26 

The model equation for the scalar energies can then be derived from (3.73) on p. 78  

26 As described in Fox (1995), the wavenumber bands are chosen to be as large as possible, subject to die condition that the characteristic time scales decrease as the band numbers increase. This condition is needed to ensure that scalar energy does not pile up at intermediate wavenumber bands. The rate-controlling step in equilibrium spectral decay is then die scalar spectral energy transfer rate (T ) from die lowest wavenumber band. 

28 ku is defined to be die same as kdi from die model energy spectrum in Chapter 2. Thus, it lies approximately one decade below die Kolmogorov wavenumber. 

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A transported PDF extension of the Hamelet model can be derived in a similar manner using the Lagrangian spectral relaxation model (Fox 1999) for the joint scalar dissipation rate. 

The spectral relaxation model of the scalar dissipation rate in homogeneous turbulence. Physics of Fluids 7, 1082-1094. 

PA at l. 48 eV appeal s instantaneously, shows spectral relaxation to the red, and decays on the same timescale of SE, as shown in Figure 8-9. We assign the observed PA to singlet Bu exciton transitions towards higher lying even parity (A ) states. We can speculate on the nature of this state within the proposed model. A possible candidate for the final slate is the inirachain biexciton. However, its energy level is located below the two-exciton stale by an amount equal to the bind-... 

The concept of a T2 cut-off that partitions the relaxation time distribution between the pores which can be displaced and those that cannot does not always apply. An exception is when there is significant diffusional coupling between the micropores that retain water at a high capillary pressure and the macropores in close proximity to the microporous system [26, 27]. A spectral BVI model or a forward model has been suggested to interpret these systems [30, 31, 53]. 

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). 

Figure 4.8. Sketch of wavenumber bands in the spectral relaxation (SR) model. The scalar-dissipation wavenumber kd lies one decade below the Batchelor-scale wavenumber kb. All scalar dissipation is assumed to occur in wavenumber band [/cd, oo). Wavenumber band [0, k ) denotes the energy-containing scales. The inertial-convective sub-range falls in wavenumber bands [k, k3 ), while wavenumber bands [/c3, /cD) contain the viscous-convective sub-range.

Given a stochastic model for the turbulence frequency, it is natural to enquire how fluctuations in co will affect the scalar dissipation rate (Anselmet and Antonia 1985 Antonia and Mi 1993 Anselmet et al. 1994). In order to address this question, Fox (1997) extended the SR model discussed in Section 4.6 to account for turbulence frequency fluctuations. The resulting model is called the Lagrangian spectral relaxation (LSR) model. The LSR model has essentially the same form as the SR model, but with all variables conditioned on the current and past values of the turbulence frequency [ /(. ),. v < r. In order to simplify the notation, this conditioning is denoted by ( , e.g.,... 

Attempts have been made to calculate the cross-relaxation rate from Tb3+ to Eu3+ by using the spectral overlap model, which employs the donor emission spectrum and its overlap with the acceptor absorption spectrum. It is... 

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