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The analysis and inference about data observed over a general(continuous or discrete) time space. Usually related to stochastic processes and will probably receive better response under that tag.
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The derivation of Reynolds-averaged Navier-Stokes equations
The equation in the OP is not correct, it should read
$$\overline{u_iu_j} = \overline{(\bar{u_i}+u_i')(\bar{u_j} + u_j')} = \overline{\bar{u_i}\bar{u_j}+\bar{u_i}u_j'+u_i'\bar{u_j}+u_i'u_j'} = \bar{u_ …