Joint scalar dissipation rate conditional

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... 

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... 

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 (1) follows directly from the definitions of the joint scalar dissipation rates and the covariance matrices, i.e.,... 

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... 

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... 

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... 

Applying the same procedure to higher-dimensional mixture-fraction vectors yields expressions of the same form as (6.130). Note also that for any set of bounded scalars that can be linearly transformed to a mixture-fraction vector, (6.115) can be used to find the corresponding joint conditional scalar dissipation rate matrix starting from (e% C). 

The scalar dissipation rate acts as an external parameter that is imposed on the flamelet structure by the mixture fraction field [15]. It describes the influence of the turbulent flow field on the laminar flame stracture. Both the mixture fraction and the scalar dissipation rate fluctuate on turbulent flows, and their statistical distribution needs to be considered. If the joint pdf P Z, Xst) (where Xst is x at the stoichiometric condition) is known, the Favre mean of 7 can be obtained from... 

VV( or q from equation (48) in the flame-sheet approximation. Since knowledge of the joint or conditioned functions is practically absent [27], statistical independence is often hypothesized or else it is merely assumed, less restric-tively, that the conditioned-mean dissipation equals the unconditioned mean. A small amount of data is available on unconditioned-average rates of scalar dissipation in turbulent flows (see discussions in [83]-[86]), and additional measurements are being made. These results allow estimates of Xc to be made, even though accurate calculations are beyond current capabilities. 

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