Scalar dissipation rate model

Sanders, J. P. H. and I. Gokalp (1998). Scalar dissipation rate modelling in variable density turbulent axisymmetric jets and diffusion flames. Physics of Fluids 10, 938-948. 

The description is based on the previously defined single-particle (Lagrangian) or one-point (Eulerian) joint velocity-composition (micro-)PDF, /(r,yr). As mentioned in Section 12.4.1, in the one-point description no information on the local velocity and scalar (species concentrations, temperature,. ..) gradients and on the frequency or length scale of the fluctuations is included and the related terms require closure models. The scalar dissipation rate model has to relate the micro-mixing time to the turbulence field (see (12.2-3)), either directly or via a transport equation for the turbulence dissipation rate e. A major advantage is that the reaction rate is a point value and its behavior and mean are described exactly by a one-point PDF, even for arbitrarily complex and nonlinear reaction kinetics. 

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). 

Choosing the micromixing time in a CRE micromixing model is therefore equivalent to choosing the scalar dissipation rate in a CFD model for scalar mixing. 

As seen above, the mean chemical source term is intimately related to the PDF of the concentration fluctuations. In non-premixed flows, the rate of decay of the concentration fluctuations is controlled by the scalar dissipation rate. Thus, a critical part of any model for chemical reacting flows is a description of how molecular diffusion works to damp out... 

In other closures for the chemical source term, a model for the conditional scalar dissipation rate (e

scalar Laplacian, the conditional scalar... 

As discussed in Section 2.1, in high-Reynolds-number turbulent flows the scalar dissipation rate is equal to the rate of energy transfer through the inertial range of the turbulence energy spectrum. The usual modeling approach is thus to use a transport equation for the transfer rate instead of the detailed balance equation for the dissipation rate derived from (1.27). Nevertheless, in order to understand better the small-scale physical phenomena that determine e, we will derive its transport equation starting from (2.99). 

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. 

The first factor occurs even in homogeneous flows with two inert scalars, and is discussed in Section 3.4. The second factor is present in nearly all turbulent reacting flows with moderately fast chemistry. As discussed in Chapter 4, modeling the joint scalar dissipation rate is challenging due to the need to include all important physical processes. One starting point is its transport equation, which we derive below. 

The second unclosed term is the scalar dissipation rate e. The most widely used closure for this term is the equilibrium model (Spalding 1971 Beguier et al. 1978) ... 

A number of different authors have proposed transport models for the scalar dissipation rate in the same general scale-similarity form as (4.47) ... 

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). 

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. ... 

The only other equation needed to close the model is an expression for the scalar dissipation rate, which is found by starting from (3.81) on p. 79 (also see Appendix A) 35... 

Figure 4.12. Predictions of the SR model for Re = 90 and Sc = 1 for homogeneous scalar mixing. For these initial conditions, all scalar energy is in the third wavenumber band. Curves 1-3 and D correspond to the fraction of scalar energy in each wavenumber band /( 2). Note that the forward cascade from 3 to 4 is larger than backscatter from 3 to 2. Thus the scalar dissipation rate rises suddenly before dropping back to the steady-state value.

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-... 

Application of the SLF model thus reduces to predicting the joint PDF of the mixture fraction and the scalar dissipation rate. As noted above, in combusting flows flame extinction will depend on the value of x Thus, unlike the equilibrium-chemistry method (Section 5.4), the SLF model can account for flame extinction due to local fluctuations in the scalar dissipation rate. 

As done below for two examples, expressions can also be derived for the scalar variance starting from the model equations. For the homogeneous flow under consideration, micromixing controls the variance decay rate, and thus y can be chosen to agree with a particular model for the scalar dissipation rate. For inhomogeneous flows, the definitions of G and M(n) must be modified to avoid spurious dissipation (Fox 1998). We will discuss the extension of the model to inhomogeneous flows after looking at two simple examples. 

Then, for shorter times, the scalar dissipation rate is modeled by147... 

Note that, by construction, ( >2 = 0. For cases where no reactions occur in environment 2, )i is constant, and the transport equation for (5)2 is not needed. A separate model must be provided for the scalar dissipation rate e. ... 

A separate model must be provided for the scalar dissipation rate i-y. 

The one-point PDF contains no length-scale information so that scalar dissipation must be modeled. In particular, the mixing time scale must be related to turbulence time scales through a model for the scalar dissipation rate. 

There is no information on the instantaneous scalar dissipation rate and its coupling to the turbulence field. A transported PDF micromixing model is required to determine the effect of molecular diffusion on both the shape of the PDF and the rate of scalar-variance decay. 

H) The molecular mixing model must yield the correct joint scalar dissipation rate. 

In other words, the model should recognize differences in shape of the scalar spectra and their effect on the scalar dissipation rate. 

Couple it with a model for the joint scalar dissipation rate that predicts the correct scalar covariance matrix, including the effect of the initial scalar length-scale distribution. 

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