The spectral link for frictional drag in non-uniform turbulent soap-film flows

The spectral link provides a novel relation between the frictional drag ($f$) and the exponent $\alpha$ of the turbulent energy spectrum ($E(k) \sim k^{-\alpha}$): $f \sim \rm{Re}^{(1-\alpha)/(1+\alpha)}$, where Re is the Reynolds number. The spectral link has been verified experimentally in uniform turbulent soap-film flows where the same $\alpha$, which is either 3 (the enstrophy cascade) or 5/3 (the inverse energy cascade), prevails across the width of the flow. We perform experiments on non-uniform turbulent soap-film flows where $\alpha$ switches between 3 and 5/3 across the width of the flow. Our measurements of $f$ vs. Re are in excellent accord with the localized version of the spectral link.