Evolution of vorticity in perturbed flow in a pipe

Abstract

Flows in long pipes often include “slugs” or “puffs” of complex or chaotic patterns interspersed with regions of relative calm. A similar phenomenon can occur in boundary layers, where “turbulent spots” can exist between rather calm regions of flow, especially in regions of transition from a laminar flow to a turbulent form. The vorticity field plays a significant role in these phenomena, and it is an aim of this paper to examine this feature from a point of view that might be thought to be unusual. Some forty years ago, G. I. Taylor initiated his theory of dispersion of a contaminant in the flow in a long pipe. He showed that a combination of longitudinal (axial) convection by the flow coupled with radial diffusion by molecular (or by turbulent) action produces an effective longitudinal diffusion process, which is governed approximately by the usual one-dimensional diffusion equation. Astonishingly, the effective longitudinal diffusivity coefficient depends inversely on the molecular (or turbulent or eddy) diffusivity. This remarkable phenomenon is now known as Taylor diffusion and has been much studied both experimentally and theoretically over the intervening forty years. The object of this paper is to study the development of vorticity in the flow in a pipe, with the vorticity treated as if it were a passive contaminant on a longitudinal flow through the pipe. In general, this requires an approximation, because the interaction between the velocity and vorticity fields is ignored. But, in a particular case—namely, that of a small swirling vortex perturbation on a flow in a pipe—a theoretical construction becomes possible in which the major approximation is simply that of linearization for small perturbation amplitudes. As a result, it is found that the swirl amplitude satisfies a one-dimensional diffusion equation, just as it does for a scalar contaminant in Taylor's classical work. There is, however, a significant difference in our problem of the evolution of vorticity or swirl: the effective longitudinal diffusion coefficient, which, like Taylor's, depends inversely on the molecular diffusivity, changes sign at a particular value of the Reynolds number. Thus, for Reynolds numbers lower than the critical value, the effective longitudinal diffusion coefficient is positive; in contrast, it is negative for Reynolds numbers larger than the critical value. The latter result implies that a focusing takes place instead of dispersion. The implications of this are discussed.