  We provide a rigorous analysis of the self-similar solution of the temporal
turbulent boundary layer, recently proposed in [2], in which a body force is
used to maintain a statistically steady turbulent boundary layer with periodic
boundary conditions in the streamwise direction. We derive explicit expressions
for the forcing amplitudes which can maintain such flows, and identify those
which can hold either the displacement thickness or the momentum thickness
equal to unity. This opens the door to the first main result of the paper,
which is to prove upper bounds on skin friction for the temporal turbulent
boundary layer. We use the Constantin-Doering-Hopf bounding method to show,
rigorously, that the skin friction coefficient for periodic turbulent boundary
layer flows is bounded above by a uniform constant which decreases
asymptotically with Reynolds number. This asymptotic behaviour is within a
logarithmic correction of well-known empirical scaling laws for skin friction.
This gives the first evidence, applicable at asymptotically high Reynolds
numbers, to suggest that the self-similar solution of the temporal turbulent
boundary layer exhibits statistical similarities with canonical, spatially
evolving, boundary layers. Furthermore, we show how the identified forcing
formula implies an alternative, and simpler, numerical implementation of
periodic boundary layer flows. We give a detailed numerical study of this
scheme presenting direct numerical simulations up to Re_theta = 2000 and
implicit large-eddy simulations up to Re_theta = 8300, and show that these
results compare well with data from canonical spatially evolving boundary
layers at equivalent Reynolds numbers.
