Mean convected derivative first form

We start by considering an arbitrary measurable10 one-point11 scalar function of the random fields U and 0 Q(U, 0). Note that, based on this definition, Q is also a random field parameterized by x and t. For each realization of a turbulent flow, Q will be different, and we can define its expected value using the probability distribution for the ensemble of realizations.12 Nevertheless, the expected value of the convected derivative of Q can be expressed in terms of partial derivatives of the one-point joint velocity, composition PDF 13 

The expected value on the left-hand side is taken with respect to the entire ensemble of random fields. However, as shown for the velocity derivative starting from (2.82) on p. 45, only two-point information is required to estimate a derivative.14 The first equality then follows from the fact that the expected value and derivative operators commute. In the two integrals after the second equality, only /u, / (V, 0 x, t) depends on x and t 

10 The term measurable (Billingsley 1979) is meant to exclude poorly behaved functions for which the probability cannot be defined. In addition, we will need to exclude functions which blow up so quickly at infinity that higher-order moments are undefined. 

11 Hence, we exclude functions (or, more precisely, operators) which require multi-point information such as 2(U, / ) = U(ti,x)-U(f2,x). 

12 In other words, each member of the ensemble represents the entire time/space history of a single experiment. For a turbulent flow, each experiment will result in a different time/space history, and the infinite collection of all such experiments constitutes an ensemble. Note that one realization contains an enormous amount of information (i.e., at least as much as is produced by a DNS of a turbulent flow which saves the fields at every time step). 

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