Scalar dissipation rate joint

The transport equation for the joint scalar dissipation rate can be derived following the same steps used to derive (3.110). We begin by defining the fluctuating scalar gradients as 

We will also need to define the Jacobian matrix of the chemical source term J with components 

For example, for the second-order reaction in (3.142) the Jacobian matrix is given by 

Differentiating both sides of (1.28) with respect to x, yields the transport equation for 

The derivation of the transport equation for giagip is analogous to that used to derive the transport equation for the scalar covariance. The resultant expression is 

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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 dissipation term35 in (3.136) is written as the product of the joint scalar dissipation rate eap defined by... 

Unlike the turbulence dissipation rate tensor, which is isotropic at high Reynolds number, the joint scalar dissipation rate tensor is usually highly anisotropic. Indeed, when r< = T, it is often the case for inert scalars that eap = eaa = , so that the joint scalar dissipation rate tensor is singular. 

The physics associated with the joint scalar dissipation rate is further complicated by two additional factors ... 

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. 

We next define the fluctuating joint scalar dissipation rate rap by... 

At high Reynolds number, we again find from (3.165) and (3.166) that the joint scalar dissipation rate is proportional to the cospectral energy transfer rate, i.e.,... 

However, because fast chemical reactions often occur over very small scales, strong coupling may exist between the chemical-source-term Jacobian and the joint scalar dissipation rate. These difficulties render computational approaches based on solving the joint scalar... 

In homogeneous turbulence, the governing equations for the scalar covariance, (3.137), and the joint scalar dissipation rate, (3.166), reduce, respectively, to... 

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-... 

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... 

The first term on the right-hand side of this expression is proportional to the joint scalar dissipation rate eap (defined in (3.139), p. 90), while the second term corresponds to molecular transport in real space 46 Thus,... 

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

II) Joint scalar dissipation rate must be correct... 

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. 

Based on the above examples, we can conclude that while localness is a desirable property, it is not sufficient for ensuring physically realistic predictions. Indeed, a key ingredient that is missing in all mixing models described thus far (except the FP and EMST70 models) is a description of the conditional joint scalar dissipation rates (e ) and their dependence on the chemical source term. For example, from the theory of premixed turbulent flames, we can expect that (eY F, f) will be strongly dependent on the chemical... 

The model proposed by Fox (1999) also accounts for fluctuations in the joint scalar dissipation rate. Here we will look only at the simpler case, where the is deterministic. 

This expression does not determine the mixing model uniquely. However, by specifying that the diffusion matrix in the resulting FP equation must equal the conditional joint scalar dissipation rate,88 the FP model for the molecular mixing term in the form of (6.48)... 

This model is consistent with (6.67), and can be seen as a multi-variate version of the IEM model. The role of the second term (eC 1) is simply to compensate for the additional diffusion term in (6.91). Note that, like with the flamelet model and the conditional-moment closure discussed in Chapter 5, in the FP model the conditional joint scalar dissipation rates ( ap ip) must be provided by the user. Since these functions have many independent variables, and can be time-dependent due to the effects of transport and chemistry, specifying appropriate functional forms for general applications will be non-trivial. However, in specific cases where the scalar fields are perfectly correlated, appropriate functional forms can be readily established. We will return to this question with specific examples below. 

Property (ii) is also controlled by the behavior of Sg(0)Cg(0). In general, the diffusion matrix should have the property that it does not allow movement in the direction normal to the surface of the allowable region.100 Defining the surface unit normal vector by n(0 ), property (ii) will be satisfied if Sg(0 )Cg(0 )n(0 ) = 0, where 0 lies on the surface of the allowable region. This condition implies that (e 10 )n(0 ) = 0, which Girimaji (1992) has shown to be true for the single-scalar case. Thus, the FP model satisfies property (ii), but the user must provide the unknown conditional joint scalar dissipation rates that satisfy (e 0 )n(0+) = 0. 

Condition (2) is more difficult to satisfy, and requires that the functional form of the conditional joint scalar dissipation rates be carefully chosen. For example, one can construct a model of the form... 

Like the IEM model, the FP model weakly satisfies property (iv). Likewise, property (v) can be built into the model for the joint scalar dissipation rates (Fox 1999), and the Sc dependence in property (vi) is included explicitly in the FP model. Thus, of the three molecular mixing models discussed so far, the FP model exhibits the greatest number of desirable properties provided suitable functional forms can be found for (e 0). 

The most common choice is for the components of Z to be uncorrelated standardized Gaussian random variables. For this case, ez z) = z = diag(szj,. .., szNs), i.e., the conditional joint scalar dissipation rate matrix is constant and diagonal. 

Note that if g is invertible, then G will be full rank. The rank of (ez Z) will thus determine the rank of (e 0) and the number of linearly independent scalars.104 The conditional joint scalar dissipation rate matrix is given by105... 

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. 

More generally, by using the linear transformation given in (5.107) on p. 167, the mixing model can be decomposed into a non-premixed, inert contribution for and a premixed, 118 reacting contribution for y>rp. It may then be possible to make judicious assumptions concerning the joint scalar dissipation rate. For example, if the spatial gradients of and y>rp are assumed to be uncorrelated, then... 

Obviously, the rate of mixing (as measured by the magnitude of joint scalar dissipation rates) will be strongly affected by turbulence, and thus must be modeled separately. 

Forthe FP model, the shape information is contained in the shape matrix H(< ), and rate information is contained in die mean joint scalar dissipation rate matrix . 

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... 

In order to close (Jwe can recognize that because J(0) depends only on the 0, it is possible to replace e by (e The closure problem then reduces to finding an expression for the doubly conditioned joint scalar dissipation rate matrix. For example, if the FP model is used to describe scalar mixing, then a model of the form... 

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