Conditional fluxes the unclosed terms

The conditional fluxes play a key role in the PDF transport equation. Writing them out using the definitions of A, and a yields20 

The first term on the right-hand side of (6.20) is unclosed, and corresponds to the effects of viscous dissipation and pressure fluctuations. The closure of this term is the principal 

19 It is now obvious that the addition of other random variables simply leads to additional conditional flux terms on the right-hand side of the PDF transport equation. The major challenge is thus to find appropriate closures for the conditional fluxes. 

20 Note that, unlike unconditional expected values, conditional expected values do not usually commute with derivatives. For example, (V2U V) V2(U V) = 0, whereas (V2U) = V2(U). 

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