  The relation between the shape of the force driving a turbulent flow and the
upper bound on the dimensionless dissipation factor $\beta$ is presented. We
are interested in non-trivial (more than two wave numbers) forcing functions in
a three dimensional domain periodic in all directions. A comparative analysis
between results given by the optimization problem and the results of Direct
Numerical Simulations is performed. We report that the bound on the dissipation
factor in the case of infinite Reynolds numbers have the same qualitative
behavior as for the dissipation factor at finite Reynolds number. As predicted
by the analysis, the dissipation factor depends strongly on the force shape.
However, the optimization problem does not predict accurately the quantitative
behavior. We complete our study by analyzing the mean flow profile in relation
to the Stokes flow profile and the optimal multiplier profile shape for
different force-shapes. We observe that in our 3D-periodic domain, the mean
velocity profile and the Stokes flow profile reproduce all the characteristic
features of the force-shape. The optimal multiplier proves to be linked to the
intensity of the wave numbers of the forcing function.
