Macroscopic effects of the spectral structure in turbulent flows

There is a missing link between the macroscopic properties of turbulent flows1–4, such as the frictional drag5 of a wall-bounded flow, and the turbulent spectrum1,6,7. The turbulent spectrum is a power law of exponentα (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 flows8, where α = 3 (refs 9,10), and in pipe flows5, where α = 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 scales13 as Re−1/4. Each of these scalings may be predicted from the attendant value of α by using a new theory14–16, in which the frictional drag is explicitly linked to the turbulent spectrum