Abstract
We derive a rigorous upper bound on the energy dissipation rate in a smoothed version of plane Poiseuille flow in which a minimum length scale parallel to the walls is imposed. Due to the smaller number of degrees of freedom, much higher Reynolds numbers can be reached in numerical simulations of this system. These indicate that the turbulent energy dissipation rate achieves the upper bound scaling except for an extra logarithmic factor of
1/(
log
Re
)
2
.
Assuming that this discrepancy in scaling between the best known upper bound and true dissipation carries over to the full plane Poiseuille problem, this result presents further evidence that the viscous dissipation rate there is
O(1/(
log
Re
)
2
)
(in inertial units) and hence vanishes as the Reynolds number
Re
→∞.
The mean flow profiles which emerge from this smoothed system are strikingly similar to those which arise by adding stress-reducing additives to a wall-bounded shear flow. Both show familiar log regions with essentially the same gradient as the undisturbed (polymer free or no minimum length scale) system but which are now displaced vertically in the traditional
U
+
−z
+
diagram.