Scalar-dissipation transport equation

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

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In the most systematic application of this approach, Harlow and co-workers at Los Alamos have derived a transport equation for the full Reynolds stress tensor pu u j. They have coupled this equation with a scalar dissipation transport equation and have utilized with various semi-empirical approximations to evaluate the numerous unknown velocity, velocity-pressure, and velocity-temperature correlations which appear in the formulation. While this treatment is fairly vigorous, extensive compu-... 

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) ... 

The firsttwo terms on the right-hand side of this expression are responsible for spatial transport of scalar dissipation. In high-Reynolds-number turbulent flows, the scalar-dissipation flux (iijC ) is the dominant term. The other terms on the right-hand side are similar to the corresponding terms in the dissipation transport equation ((2.125), p. 52), and are defined as follows. 

Only for an isothermal, first-order reaction where Sa = —k a will the chemical source term in (3.102) be closed, i.e., ++(<+ = h (u,(pa). Indeed, for more complex chemistry, closure of the chemical source term in the scalar-flux transport equation is a major challenge. However, note that, unlike the scalar-flux dissipation term, which involves the correlation between gradients (and hence two-point statistical information), the chemical source term is given in terms of u(x, t) and 0(x, t). Thus, given the one-point joint velocity, composition PDF /u,chemical source term is closed, and can be computed from... 

The Reynolds-averaged approach is widely used for engineering calculations, and typically includes models such as Spalart-Allmaras, k-e and its variants, k-co, and the Reynolds stress model (RSM). The Boussinesq hypothesis, which assumes pt to be an isotropic scalar quantity, is used in the Spalart-Allmaras model, the k-s models, and the k-co models. The advantage of this approach is the relatively low computational cost associated with the computation of the turbulent viscosity, fit. For the Spalart-Allmaras model, one additional transport equation representing turbulent viscosity is solved. In the case of the k-e and k-co models, two additional transport equations for the turbulence kinetic energy, k, and either the turbulence dissipation rate, s, or the specific dissipation rate, co, are solved, and pt is computed as a function of k and either e or co. Alternatively, in the RSM approach, transport equations can be solved for each of the terms in the Reynolds stress tensor. An additional scale-determining equation (usually for s) is also required. This means that seven additional transport equations must be solved in 3D flows. 

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). 

In developing closures for the chemical source term and the PDF transport equation, we will also come across conditional moments of the derivatives of a field conditioned on the value of the field. For example, in conditional-moment closures, we must provide a functional form for the scalar dissipation rate conditioned on the mixture fraction, i.e.,... 

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)... 

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. 

The transport equation for the scalar dissipation rate of an inert scalar can be derived starting from (3.90). We begin by defining the fluctuating scalar gradient as... 

Multiplying (3.111) by 2Y and Reynolds averaging yields the final form for the scalar-dissipation-rate transport equation ... 

Using (3.130), the transport equation for the scalar dissipation rate in high-Reynolds-number homogeneous turbulence becomes... 

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. 

On the other hand, on the bounding hypersurfaces the normal diffusive flux must be null. However, this condition will result naturally from the fact that the conditional joint scalar dissipation rate must be zero-flux in the normal direction on the bounding hypersurfaces in order to satisfy the transport equation for the mixture-fraction PDF.122... 

Comparing (5.377) with (3.105) on p. 85 in the high-Reynolds-number limit (and with e = 0), it can be seen that (5.378) is a spurious dissipation term.149 This model artifact results from the presumed form of the joint composition PDF. Indeed, in a transported PDF description of inhomogeneous scalar mixing, the scalar PDF relaxes to a continuous (Gaussian) form. Although this relaxation process cannot be represented exactly by a finite number of delta functions, Gs and M1 1 can be chosen to eliminate the spurious dissipation term in the mixture-fraction-variance transport equation.150... 

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. ... 

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