Reynolds turbulence decomposition approach

We assume that is a velocity fluctuation in the x direction only and v in the y direction only. The overbar denotes time average, so that by definition, the averages of v, and p flucttia iK are zero (stochastically random), and the partial derivatives in time of the mean quantities U,V,P are zeros (the Reynolds turbulence decomposition approach, according to which velocities and pressures can be decomposed to time-dependent and time-independent components). 

As of this writing, the only practical approach to solving turbulent flow problems is to use statistically averaged equations governing mean flow quantities. These equations, which are usually referred to as the Reynolds equations of motion, are derived by Reynold s decomposition of the Navier-Stokes equations (18). The randomly changing variables are represented by a time mean and a fluctuating part ... 

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