Turbulence Theory: Bounds and Higher-Order Statistics

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The objective of this research is the development of turbulence approximations which provide upper and lower bounds on quantities of physical interest. This work is in the spirit of the bounding theory of Howard and Busse but proceeds from a decimation scheme in which only a subset of the modes of the flow system are treated explicitly. The remaining, implicit modes are represented by stochastic forcing terms which are subject to an in infinite set of moment constraints taken from the exact dynamics. To find bounds, it is proposed to seek solutions which extremalize the chosen quantities under a restricted subset of these constraints, imposed together with realizability inequalities. Since the constraints are exact, bounds so constructed would be rigorous. Fourth.order statistics play an essential role in the development of bounds. These statistics are additionally important because of recent Navier.Stokes simulation studies which show interesting behavior of mean.square helicity and the mean square of the total nonlinear term. It is proposed to study 4th.order statistics by the decimation scheme. The complexity of the work can be reduced by using integral rather than detailed forms of 4th.order moment constraints.