The exact evolution of the scalar variance in pipe and channel flow

Abstract

In 1953 G.I. Taylor showed theoretically and experimentally that a passive tracer diffusing in the presence of laminar pipe flow would experience an enhanced diffusion in the longitudinal direction beyond the bare molecular diffusivity K in the amount, where a is the pipe radius and U is the maximum fluid velocity. This behavior is predicted to arise after a transient timescale a2 k, the diffusive timescale for the tracer to cross the pipe. Typically, U is very small, so provided a fairly long time has passed, this is a very large diffusive boost. Before this timescale, the evolution is expected to be anomalous, meaning the scalar variance does not grow linearly in time. A few attempts to compute this anomalous growth have been made in the literature for different special cases with different approximations. Here, we derive an exact approach which provides the scalar variance evolution valid for all times for channel and pipe flow for the case of vanishing Neumann boundary conditions. We show how this formula limits to the Taylor regime, and rigorously study the anomalous regime for a range of initial data. We find that the anomalous timescales and exponents depend strongly upon the form of the data. For initial data whose transverse variation is a delta function on the centerline, the anomalous regime emerges after a timescale, 1/3, with variance growing as tα, with α=4. In contrast, for the case of uniform data (independent of the transverse variable), the anomalous timescale is K/U2, with exponent α=2, and this result is generalized for generic shear flows given that the initial condition is not a transverse Dirac delta function. Further, these exact formulas explicitly show what features the short time approximations which ignore physical boundaries are able to capture. © 2010 International Press.