Predicting the enhancement of mixing-driven reactions in nonuniform flows using measures of flow topology

The ability for reactive constituents to mix is often the key limiting factor for the completion of reactions across a huge range of scales in a variety of media. In flowing systems, deformation and shear enhance mixing by bringing constituents into closer proximity, thus increasing reaction potential. Accurately quantifying this enhanced mixing is key to predicting reactions and typically is done by observing or simulating scalar transport. To eliminate this computationally expensive step, we use a Lagrangian stochastic framework to derive the enhancement to reaction potential by calculating the collocation probability of particle pairs in a heterogeneous flow field accounting for deformations. We relate the enhanced reaction potential to three well known flow topology metrics and demonstrate that it is best correlated to (and asymptotically linear with) one: the largest eigenvalue of the (right) Cauchy-Green tensor.

Figure 1

Magnitude of the velocity field and pressure equipotentials (dark solid lines). Flow is from left to right and velocities are in $m/d$.

Figure 2

(a) Square root of the Okubo-Weiss parameter $\sqrt{\Theta }$ and (b) finite-time Lyapunov exponent. The complex nature of the velocity field causes significant noise in $\Theta$.

Figure 3

(a) Largest eigenvalue of the Cauchy-Green tensor ${\lambda }_{\mathbf{C}}$ and (b) relative enhancement of the CLD ${\nu }_E\left(\mathbf{s}\right)$. The areas with the highest changes in CLD are predicted very accurately from ${\lambda }_{\mathbf{C}}$ (Table 1).