We present a numerical strategy that allows us to explore the full scope of the Doering–Constantin variational principle for computing rigorous upper bounds on energy dissipation in turbulent shear flow. The key is the reformulation of this principle's spectral constraint as a boundary value problem that can be solved efficiently for all Reynolds numbers of practical interest. We state results obtained for the plane Couette flow, and investigate in detail a simplified model problem that can serve as a definite guide for the application of the variational principle to other flows. The most notable findings are a bifurcation of the minimizing wavenumber and a pronounced minimum of the bound at intermediate Reynolds numbers, and a distinct asymptotic scaling of the optimized variational parameters.