Macroscopic effects of the spectral structure in turbulent flows*

There is a missing link between macroscopic properties of turbulent flows, such as the frictional drag of a wall-bounded flow, and the turbulent spectrum. To seek the missing link we carry out unprecedented experimental measurements of the frictional drag in turbulent soap-film flows over smooth walls. These flows are effectively two-dimensional, and we are able to create soap-film flows with the two types of turbulent spectrum that are theoretically possible in two dimensions: the ``enstrophy cascade,'' for which the spectral exponent $\alpha = 3$, and the ``inverse energy cascade,'' for which the spectral exponent $\alpha = 5/3$. We find that the functional relation between the frictional drag $f $ and the Reynolds number $\rm{Re}$ depends on the spectral exponent: where $\alpha = 3$, $f \sim Re^{-1/2}$; where $\alpha = 5/3$, $f \sim Re^{-1/4}$. Each of these scalings may be predicted from the attendant value of $\alpha$ by using a recently proposed spectral theory of the frictional drag. In this theory the frictional drag of turbulent flows on smooth walls is predicted to be $f \sim Re^{(1-\alpha)/(1+\alpha)}$.