Scalar-flux transport equation

Like the Reynolds stresses, the scalar flux obeys a transport equation that was derived in Section 3.3  

In analogy to (4.55), the turbulent flux terms in (4.78) are usually modeled by invoking a gradient-diffusion hypothesis 21 

however, comparison with data (Launder 1996) indicates that the additional complexity does not yield much improvement over (4.79). 

It can be noted that this model is not symmetric with respect to interchanging i and j, although the unclosed term is symmetric. 

As for the models derived from the PDF transport equation, nearly all widely used models for Ilf can be expressed as 

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The derivation of the scalar-flux transport equation proceeds in exactly the same manner as with the Reynolds stresses. We first multiply (2.99), p. 48, by

[Pg.101]

This type of model is usually referred to as an algebraic scalar-flux model. Similarmodels for the Reynolds-stress tensor are referred to as algebraic second-moment (ASM) closures. They can be derived from the scalar-flux transport equation by ignoring time-dependent and spatial-transport terms. 

At high Reynolds numbers, the modeled scalar-flux transport equation thus reduces (using (4.78)) to... 

We shall see that a conditional acceleration model in the form of (6.48) is equivalent to a stochastic Lagrangian model for the velocity fluctuations whose characteristic correlation time is proportional to e/k. As discussed below, this implies that the scalar flux (u,

joint velocity, composition PDF level, and thus that a consistent scalar-flux transport equation can be derived from the PDF transport equation. 

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... 

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