SRS models

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.

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

Reynolds numbers, its value is significantly smaller than the high-Reynolds-number limit. Despite its inability to capture low-Reynolds-number effects on the steady-state scalar dissipation rate, the SR model does account for Reynolds-number and Schmidt-number effects on the dynamic behavior of R(t). 

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... 

The terms involving y in the SR model equations correspond to the fraction of the scalar-variance production that falls into a particular wavenumber band. In principle, yn could be found from the scalar-flux spectrum (Fox 1999). Instead, it is convenient to use a self-similarity hypothesis that states that for Sc = 1 at spectral equilibrium the fraction of scalar variance that lies in a particular wavenumber band will be independent of V. Applying this hypothesis to (4.103)-(4.106) yields 31... 

In the definition of b, Ro is the equilibrium mechanical-to-scalar time-scale ratio found with Sc = 1 and = 0.34 The parameters Cd, Cb, and Cd appear in the SR model for the scalar dissipation rate discussed below. Note that, by definition, xi + Yi + K3 + Kd = 1. ... 

Thus the SR model yields the standard scalar-variance transport equation for homogeneous flow ... 

The initial conditions for (

model predictions with four different initial conditions are shown in... 

Figure 4.10. Predictions of the SR model for Re, =90 and Sc = 1 for homogeneous scalar mixing in stationary turbulence. For these initial conditions, all scalar energy is in the first wavenumber band. Curves 1-3 and D correspond to the fraction of scalar energy in each wavenumber band < 2)n/ 4> 2 - Note the relatively long transient period needed for R(t) to approach its asymptotic value of A o = 2.

Figs. 4.10-4.13. In stationary turbulence (k and e constant), the SR model attains a dynamical steady state wherein the variables (/2) and e reach their equilibrium values. In this limit (for Sc = 1),... 

The SR model can be extended to inhomogeneous flows (Tsai and Fox 1996a), and a Lagrangian PDF version (LSR model) has been developed and validated against DNS data (Fox 1997 Vedula et al. 2001). We will return to the LSR model in Section 6.10. 

At present, there exists no completely general RANS model for differential diffusion. Note, however, that because it solves (4.37) directly, the linear-eddy model discussed in Section 4.3 can describe differential diffusion (Kerstein 1990 Kerstein et al. 1995). Likewise, the laminar flamelet model discussed in Section 5.7 can be applied to describe differential diffusion in flames (Pitsch and Peters 1998). Here, in order to understand the underlying physics, we will restrict our attention to a multi-variate version of the SR model for inert scalars (Fox 1999). 

The SR model introduced in Section 4.6 describes length-scale effects and contains an explicit dependence on Sc. In this section, we extend the SR model to describe differential diffusion (Fox 1999). The key extension is the inclusion of a model for the scalar covariance (4> aft p) and the joint scalar dissipation rate. In homogeneous turbulence, the covari-... 

A corresponding Schmidt number is defined by Sc, = v/Tap. Then, when defining the SR model for the covariance, we will simply replace Sc by Scap (e.g., in (4.98),... 

Like the scalar variance, (4.102), in the multi-variate SR model the scalar covariance is divided into finite wavenumber bands (

[Pg.155]

In summary, the multi-variate SR model is found by applying the uni-variate SR model to each component of Wa4> p)n and eap. For the case where ra = I, the model equations for all components will be identical. The model predictions then depend only on the initial conditions, which need not be identical for each component. In order to see how differential... 

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. 

In order to illustrate how the multi-variate SR model works, we consider a case with constant Re>. = 90 and Schmidt number pair Sc = (1, 1/8). If we assume that the scalar fields are initially uncorrelated (i.e., pup 0) = 0), then the model can be used to predict the transient behavior of the correlation coefficients (e.g., pap(i)). Plots of the correlation coefficients without (cb = 0) and with backscatter (Cb = 1) are shown in Figs. 4.14 and 4.15, respectively. As expected from (3.183), the scalar-gradient correlation coefficient gap(t) approaches l/yap = 0.629 for large t in both figures. On the other hand, the steady-state value of scalar correlation pap depends on the value of Cb. For the case with no backscatter, the effects of differential diffusion are confined to the small scales (i.e., (), / h and s)d) and, because these scales contain a relatively small amount of the scalar energy, the steady-state value of pap is close to unity. In contrast, for the case with backscatter, de-correlation is transported back to the large scales, resulting in a lower steady-state value for p p. 

We will next consider the case of decaying scalars where V p = 0. For this case, it is convenient to assume that the scalars are initially perfectly correlated so that pap(0) = gap(0) = pd(0) = 1. The multi-variate SR model can then be used to describe how the scalars de-correlate with time. We will again consider a case with constant Rc>, = 90 and Schmidt number pair Sc = (1, 1/8). 

Typical model predictions without and with backscatter are shown in Figs. 4.16 and 4.17, respectively. It can be noted that for decaying scalars the effect of backscatter on de-correlation is dramatic. For the case without backscatter (Fig. 4.16), after a short transient period the correlation coefficients all approach steady-state values. In contrast, when backscatter is included (Fig. 4.17), the correlation coefficients slowly approach zero. The rate of long-time de-correlation in the multi-variate SR model is thus proportional to the backscatter constant Cb-... 

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 extension of the SR model to differential diffusion is outlined in Section 4.7. In an analogous fashion, the LSR model can be used to model scalars with different molecular diffusivities (Fox 1999). The principal changes are the introduction of the conditional scalar covariances in each wavenumber band 4> a4> p) and the conditional joint scalar dissipation rate matrix (e). For example, for a two-scalar problem, the LSR model involves three covariance components (/2, W V/A) and and three joint dissipation... 

Gap is the corresponding scalar-covariance source term, and Tap is the scalar-covariance transfer spectrum. In the following, we will relate the SR model for the scalar variance to (A.2) however, analogous expressions can be derived for the scalar covariance from (A.4) by following the same procedure. 

A key assumption in deriving the SR model (as well as earlier spectral models see Batchelor (1959), Saffman (1963), Kraichnan (1968), and Kraichnan (1974)) is that the transfer spectrum is a linear operator with respect to the scalar spectrum (e.g., a linear convection-diffusion model) which has a characteristic time constant that depends only on the velocity spectrum. The linearity assumption (which is consistent with the linear form of (A.l)) ensures not only that the scalar transfer spectra are conservative, but also that if Scap = Scr in (A.4), then Eap ic, t) = Eyy k, t) for all t when it is true for t = 0. In the SR model, the linearity assumption implies that the forward and backscatter rate constants (defined below) have the same form for both the variance and covariance spectra, and that for the covariance spectrum the rate constants depend on the molecular diffusivities only through Scap (i.e., not independently on Sc or Sep). 

The SR model can be derived from (A.2) using the cut-off wavenumbers for each wavenumber band. The Kolmogorov wavenumber is defined by... 

In the present version of the SR model, the fractions y, and yn are assumed to be time-independent functions of Rei and Sc. Likewise, the scalar-variance source term Va is closed with a gradient-diffusion model. The SR model could thus be further refined (with increased computational expense) by including an explicit model for the scalar-flux spectrum. 

In the SR model, ku is chosen such that eau ea, i.e., so that the bulk of the scalar dissipation occurs in the final wavenumber band. Thus, the scalar dissipation terms appearing... 

Note that, unlike with the continuous representation (A.2), the discrete representation used in the SR model requires an explicit model for the scalar dissipation rate ea. 

In the SR model, forward and backscatter rate constants are employed to model the spectral transport terms ... 

P(j+i)j for / = 1— 1 to be independent of Sc. This is the assumption employed in the SR model, but it can be validated (and modified) using DNS data for the scalar spectrum and the scalar-scalar transfer function. The linearity assumption discussed earlier implies that the rate constants will be unchanged (for the same Reynolds and Schmidt numbers) when they are computed using the scalar-covariance transfer spectrum. 

Note that at spectral equilibrium the integral in (A.33) will be constant and proportional to ea (i.e., the scalar spectral energy transfer rate in the inertial-convective sub-range will be constant). The forward rate constants a j will thus depend on the chosen cut-off wavenumbers through their effect on (computationally efficient spectral model possible, the total number of wavenumber bands is minimized subject to the condition that... 

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