Macroscopic effects of the spectral structure in turbulent flows

There is a missing link between the macroscopic properties of turbulent flows(1-4), such as the frictional drag(5) of a wall-bounded flow, and the turbulent spectrum(1,6,7). The turbulent spectrum is a power law of exponent alpha (the 'spectral exponent') that gives the characteristic velocity of a turbulent fluctuation (or 'eddy') of size s as a function of s (ref. 1). Here we seek the missing link by comparing the frictional drag in soap-film flows(8), where alpha = 3 (refs 9,10), and in pipe flows(5), where alpha = 5/3 (refs 11,12). For moderate values of the Reynolds number Re, we find experimentally that in soap-film flows the frictional drag scales as Re-1/2, whereas in pipe flows the frictional drag scales(13) as Re-1/4. Each of these scalings may be predicted from the attendant value of a by using a new theory(14-16), in which the frictional drag is explicitly linked to the turbulent spectrum