Squaring the Circle: Geometric Skewness and Symmetry Breaking for Passive Scalar Transport in Ducts and Pipes

We study the role geometry plays in the emergence of asymmetries in diffusing passive scalars advected by pressure-driven flows in ducts and pipes of different aspect ratios. We uncover nonintuitive, multi-time-scale behavior gauged by a new statistic, which we term "geometric skewness" SG, which measures instantaneously forming asymmetries at short times due to flow geometry. This signature distinguishes elliptical pipes of any aspect ratio, for which S-G = 0, from rectangular ducts whose SG is generically nonzero, and, interestingly, shows that a special duct of aspect ratio approximate to 0.53335 behaves like a circular pipe as its geometric skewness vanishes. Using a combination of exact solutions, novel short-time asymptotics, and Monte Carlo simulations, we establish the relevant time scales for plateaus and extrema in the evolution of the skewness and kurtosis for our class of geometries. For ducts limiting to channel geometries, we present new exact, single-series formulas for the first four moments on slices used to benchmark Monte Carlo simulations.