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” ${\mathcal{S}}^{\mathcal{G}}$, which measures instantaneously forming asymmetries at short times due to flow geometry. This signature distinguishes elliptical pipes of any aspect ratio, for which ${\mathcal{S}}^{\mathcal{G}}=0$, from rectangular ducts whose ${\mathcal{S}}^{\mathcal{G}}$ is generically nonzero, and, interestingly, shows that a special duct of aspect ratio $\approx 0.53335$ 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.

Figure 1

Channel full skewness and kurtosis (inset) exact solutions versus Monte Carlo computations, $\text{Pe}={10}^4$.

Figure 2

Mean-zero flow profile $u(y,z)$ and full skewness (left) for duct (aspect ratio $\lambda =0.2$, $\text{Pe}={10}^4$), with snapshots of the $xy$ and $xz$ projections of a sample path at times marked by red lines in the skewness graph: $\tau ={10}^{-4}$ (center) and $\tau =4.4$ (right).

Figure 3

Evolution of the full skewness for duct (pipe, inset), $\lambda =1/5$, Péclet ${10}^0–{10}^6$. The X’s denote large Pe prediction of minima.

Figure 4

Left: Full skewness evolution for ducts, $\lambda =0.01–1.0$, $\text{Pe}={10}^4$, short-time asymptotics (solid, vertical asymptote due to vanishing ${\mathcal{M}}_2$), Monte Carlo simulations (symbols). The X’s denote asymptotic prediction of minima. Center: Duct ${\mathcal{S}}^{\mathcal{G}}$ vs $\lambda$, predicting first plateaus. Inset: Duct simulation, plug $f(x)$, $\sigma \approx 0.115$. Right: Ellipse full skewness, same parameters.