Reformulation of the statistical equations for turbulent shear flow

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Abstract
Recent experimental results on the intermittent generation of shear turbulence raise questions about the significance of Reynolds averaging. In this paper, statistical equations are derived by progressively averaging the Navier-Stokes equation over a series of increasing time periods. Averaging over the shortest time smooths out part of the field (which corresponds to the highest-frequency fluctuations). The mean effect of these fluctuations may be calculated from the time-averaged equation of motion, and so eliminated from the equation describing the rest of the velocity field (that is, the unaveraged part). An iterative process leads to equations for the mean and covariance of the fluctuating field, in which the Reynolds stresses do not appear explicitly. They are represented in each cycle of the iteration by the sum of the following: (1) a constitutive relation, expressing the mean-square fluctuation in terms of the mean rate of strain, and (2) the unaveraged portion of the nonlinear term. The method resembles the renormalization group (it differs insofar as the averaging process is the defining operation). A renormalization-group analysis is used to investigate the iteration process. With some simplifying assumptions (e.g., the fluctuations are taken to be isotropic and the spectrum to be a power law), a recursion relation for the viscosity is found to reach a fixed point. In the limit of long averaging times, the mean-field equation reduces to the Reynolds equation, with the turbulent stresses replaced by an effective viscosity.