  We use momentum transfer arguments to predict the friction factor $f$ in
two-dimensional turbulent soap-film flows with rough boundaries (an analogue of
three-dimensional pipe flow) as a function of Reynolds number Re and roughness
$r$, considering separately the inverse energy cascade and the forward
enstrophy cascade. At intermediate Re, we predict a Blasius-like friction
factor scaling of $f\propto\textrm{Re}^{-1/2}$ in flows dominated by the
enstrophy cascade, distinct from the energy cascade scaling of
$\textrm{Re}^{-1/4}$. For large Re, $f \sim r$ in the enstrophy-dominated case.
We use conformal map techniques to perform direct numerical simulations that
are in satisfactory agreement with theory, and exhibit data collapse scaling of
roughness-induced criticality, previously shown to arise in the 3D pipe data of
Nikuradse.
