The rate of viscous energy dissipation in a shear layer of incompressible Newtonian fluid with injection and suction is studied by means of exact solutions, nonlinear and linearized stability theory, and rigorous upper bounds. For sufficiently large values of the suction rate, a steady laminar flow is absolutely stable at all shear rates. For sufficiently small but nonzero suction, however, the laminar flow is linearly unstable at high Reynolds numbers. We find that the rigorous upper bound on the energy dissipation rate---valid even for turbulent (and weak) solutions of the Navier-Stokes equations---scale precisely the same as the dissipation in the laminar solution in the zero viscosity limit. Both the laminar and any possible turbulent flows display a finite nonvanishing residual dissipation as the viscosity is decreased. This is a manifestation of so-called Kolmogorov-type scaling in which the energy dissipation rate becomes independent of the viscosity at high Reynolds numbers. This result establishes the sharpness of the upper bound's scaling in the vanishing viscosity limit---for these boundary conditions. It also provides a mathematical illustration of the delicacy of corrections to high Reynolds number scaling (such as the logarithmic terms as appearing in the Prandtl-von Karman "law of the wall") to perturbation in the boundary conditions. This is joint work with Edward A. Spiegel (Columbia University) and Rodney A. Worthing (University of Michigan).