Positive- and negative-effective-viscosity phenomena in isotropic and anisotropic Beltrami flows

A field-theoretic approach, analogous to Kraichnans direct-interaction approximation, to the stability theory of complex three-dimensional flows is developed. The long-wavelength stability of a class of Beltrami flows in an unbounded, viscous fluid is considered. We examine two flows in detail, to illustrate the effects of strong isotropy versus strong anisotropy in the basic flow. The effect of the small-scale flow on the long-wavelength perturbations may be interpreted as an effective viscosity. Using diagrammatic techniques, we construct the first-order smoothing and direct-interaction approximations for the perturbation dynamics. It is argued that the effective viscosity for the isotropic flow is always positive, and approaches a value independent of the molecular viscosity in the high-Reynolds-number limit; this flow is thus stable to long-wavelength disturbances. The anisotropic flow has negative effective viscosity for some orientations of the disturbance, and is therefore unstable, when its Reynolds number exceeds 2.