Scalar dissipation rate spectral model

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). 

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) ... 

A spectral model similar to (3.82) can be derived from (3.75) for the joint scalar dissipation rate eap defined by (3.139), p. 90. We will use these models in Section 3.4 to understand the importance of spectral transport in determining differential-diffusion effects. As we shall see in the next section, the spectral interpretation of scalar energy transport has important ramifications on the transport equations for one-point scalar statistics for inhomogeneous turbulent mixing. 

As discussed in Chapter 4, the modeling of the scalar dissipation rate in (3.105) is challenging due to the need to describe both equilibrium and non-equilibrium spectral... 

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. 

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. 

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.,... 

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

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. 

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