Bulk flow scaling for turbulent channel and pipe flows

We report a theory deriving bulk flow scaling for canonical wall-bounded flows. The theory accounts for the symmetries of boundary geometry (flat plate channel versus circular pipe) by a variational calculation for a large-scale energy length, which characterizes its bulk flow scaling by a simple exponent, i.e. $m=4$ for channel and 5 for pipe. The predicted mean velocity shows excellent agreement with several dozen sets of quality empirical data for a wide range of the Reynolds number (Re), with a universal bulk flow constant $\kappa\approx0.45$. Predictions for dissipation and turbulent transport in the bulk flow are also given, awaiting data verification.