We examine the background flow variational principle for calculating bounds
on the energy dissipation rate ill turbulent shear flow, and suggest to se
lect this principle's test functions such that they comply with the small-s
cale smoothness of real turbulent velocity fields. A self-consistent algori
thm implementing this requirement then yields an upper bound on the dimensi
onless dissipation coefficient which shows a weak power-law decrease at hig
h Reynolds numbers, instead of approaching a nonzero constant, as it did in
previous estimates.