  We consider two-dimensional homogeneous shear turbulence within the context
of optimal control, a multi-scale turbulence model containing the fluctuation
velocity and pressure correlations up to the fourth order; The model is
formulated on the basis of the Navier-Stokes equations, Reynolds average, the
constraints of inequality from both physical and mathematical considerations,
the turbulent energy density as the objective to be maximized, and the fourth
order correlations as the control variables. Without imposing the maximization
and the constraints, the resultant equations of motion in the Fourier wave
number space are formally solved to obtain the transient state solutions, the
asymptotic state solutions and the evolution of a transient toward an
asymptotic under certain conditions. The asymptotic state solutions are
characterized by the dimensionless exponential time rate of growth $2\sigma$
which has an upper bound of $2\sigma_{\max}$ = 0; The asymptotic solutions can
be obtained from a linear objective convex programming. For the asymptotic
state solutions of the reduced model containing the correlations up to the
third order, the optimal control problem reduces to linear programming with the
primary component of the third order correlations or a related integral
quantity as the control variable; the supports of the second and third order
correlations are estimated for the sake of numerical simulation; the existence
of feasible solutions is demonstrated when the related quantity is the control
variable. The relevance of the formulation to flow stability analysis is
suggested.
