In recent years there has been vigorous debate whether asymptotic transverse macrodispersion exists in steady three‐dimensional (3D) groundwater flows in the purely advective limit. This question is tied to the topology of 3D flow paths (termed the Lagrangian kinematics), specifically whether streamlines can undergo braiding motions or can wander freely in the transverse direction. In this study we determine which Darcy flows do admit asymptotic transverse macrodispersion for purely advective transport on the basis of the conductivity structure. We prove that porous media with smooth, locally isotropic hydraulic conductivity exhibit zero transverse macrodispersion under pure advection due to constraints on the Lagrangian kinematics of these flows, whereas either non‐smooth or locally anisotropic conductivity fields can generate transverse macrodispersion. This has implications for upscaling locally isotropic porous media to the block scale as this can result in a locally anisotropic conductivity, leading to non‐zero macrodispersion at the block scale that is spurious in that it does not arise for the fully resolved Darcy scale flow. We also show that conventional numerical methods for computation of particle trajectories do not explicitly preserve the kinematic constraints associated with locally isotropic Darcy flow, and propose a novel psuedo‐symplectic method that preserves these constraints. These results provide insights into the mechanisms that govern transverse macrodispersion in groundwater flow, and unify seemingly contradictory results in the literature in a consistent framework. These insights call into question the ability of smooth, locally isotropic conductivity fields to represent flow and transport in real heterogeneous porous media.
We analyze the dynamics of solute mixing in a vortex flow. The transport of a passive tracer is considered in a Rankine vortex. The action of a shear flow, in general, is to achieve stretching of fluid elements. A vortex flow exhibits stretching and folding of fluid elements in a way which brings adjacent fluid elements closer every turn. A strong stretching along the direction of rotation is accompanied by a concomitant thinning in the radial direction leading to a strong diffusive flux which may cause material from neighbouring regions of the mixing interface to aggregate. Through a Lagrangian concentration evolution technique, the diffusive strip method, we obtain the concentration field and pinpoint the signature of coalescence of two neighbouring concentration regions by analyzing the concentration distribution profiles. We link coalescence with reactivity for mixing-limited reactive flows. The analysis is useful to understand scalar dispersion in vortical flow structures.
We analyze the dynamics of solute mixing and reaction in a mixing-limited reactive flow by considering the transport of a tracer in a linear shear flow and in a Rankine vortex. The action of a shear flow, in general, achieves stretching of fluid elements due to the heterogeneous nature of the flow. A vortex flow exhibits not only stretching but also folding of fluid elements in a way that brings adjacent fluid elements closer at every turn. A strong stretching along the tangential direction is accompanied by a concomitant thinning in the radial direction leading to a strong diffusive flux, which may cause the material from neighboring regions of the mixing interface to aggregate. Through a Lagrangian concentration evolution technique, the diffusive strip method, we obtain the concentration field and pinpoint the signature of coalescence of two neighboring concentration regions by analyzing the concentration distribution profiles. The role of substrate deformation on the reaction kinetics of a classical heterogeneous chemical reaction is also studied where we derive analytical expressions for the coupling between the rate of product formation and the Péclet number in different time limits. Finally, the impact of coalescence on reaction rates is studied for a Rankine vortex, a result that holds important implications for simple bimolecular reactions. This analysis is useful to understand scalar dispersion in vortical flow structures and the consequences of stretching-enhanced diffusion in mixing-limited reactive flows.
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