Turbulent drag reduction in one and two dimensions

Drag reduction is investigated in one- and two-dimensional turbulent flows that are stationary and homogeneous. In the two-dimensional case, the polymer chains are deformed though advected passively through the Kraichnan cascades within a scaling analysis. The typical rate of shear must then be larger than the time of deformation of a chain. Ultimately, elastic forces compete with Reynolds stresses at an elastic cut-off similar to that defined in the Tabor–de Gennes scenario in three dimensions. There are several regimes in two dimensions because there are two cascades. In the one-dimensional case, a Burgers-type equation is coupled to a frame-indifferent equation for the viscoelastic Maxwell stress together with a stochastic force. Dissipation arises within viscoelastic shocks. The (effectively longitudinal) speed of the elastic waves is a cut-off for the shocks. This elastic cut-off shows up in velocity correlations. The implications for soap film and wire experiments are discussed.