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We explore incompressible homogeneous isotropic turbulence within the(fourth-order model) formulation of optimal control and optimization, incontrast to the classical works of Proudman and Reid (1954) and Tatsumi (1957),with the intention to fix specially their defect of negative energy spectrumvalues being developed and to examine generally the conventional closureschemes. The isotropic forms for the general and spatially degenerated fourthorder correlations of fluctuating velocity are obtained and the correspondingprimary dynamical equations are derived. The degenerated fourth ordercorrelation contains four scalar functions $D_i$, $i=1,2,3,4$, whosedetermination is the focus of closure. We discuss the constraints of equalityfor these functions as required by the self-consistency of the definition ofthe degenerated. Furthermore, we develop the constraints of inequality for thescalar functions based on the application of the Cauchy-Schwarz inequality, thenon-negativity of the variance of products, and the non-negativity of theturbulent energy spectrum. We intend to indicate the difficulty for aconventional scheme to satisfy all these constraints. As an alternative, weemploy the turbulent energy per unit volume as the objective function to bemaximized, under the constraints and the dynamical equations, with the fourscalar functions as the control variables, which is a second-order coneprogramming problem. We then treat the asymptotic state solutions at large timeand focus especially on the sub-model where the third order correlation istaken as the control variable, considering the computing resources available.

Multi-scale turbulence modeling and maximum information principle. Part 4