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 α = 3, and the “inverse energy cascade, ” for which the spectral exponent α = 5/3. We find that the functional relation between the frictional drag f and the Reynolds number Re depends on the spectral exponent: where α = 3, f ∝ Re−1/2; where α = 5/3, f ∝ Re−1/4. These findings cannot be reconciled with the classic theory of the frictional drag. The classic theory provides no means of distinguishing between one type of turbulent spectrum and another, and cannot account for the existence of a “spectral link ” between the frictional drag and the turbulent spectrum. In view of our experimental results, we conclude that the classic theory must be considered incomplete. In contrast, our findings are consistent with a recently proposed spectral theory of the frictional drag. In this theory the frictional drag of turbulent flows on smooth walls is pre-

belowground functioning of tropical biomes

RECHERCHES Elastographie par ondes de cisaillement