Modeling preasymptotic transport in flows with significant inertial and trapping effects – The importance of velocity correlations and a spatial Markov model

We study solute transport in a periodic channel with a sinusoidal wavy boundary when inertial flow effects are sufficiently large to be important, but do not give rise to turbulence. This configuration and setup are known to result in large recirculation zones that can act as traps for solutes; these traps can significantly affect dispersion of the solute as it moves through the domain. Previous studies have considered the effect of inertia on asymptotic dispersion in such geometries. Here we develop an effective spatial Markov model that aims to describe transport all the way from preasymptotic to asymptotic times. In particular we demonstrate that correlation effects must be included in such an effective model when Péclet numbers are larger than O(100) in order to reliably predict observed breakthrough curves and the temporal evolution of second centered moments. For smaller Péclet numbers correlation effects, while present, are weak and do not appear to play a significant role. For many systems of practical interest, if Reynolds numbers are large, it may be typical that Péclet numbers are large also given that Schmidt numbers for typical fluids and solutes can vary between 1 and 500. This suggests that when Reynolds numbers are large, any effective theories of transport should incorporate correlation as part of the upscaling procedure, which many conventional approaches currently do not do. We define a novel parameter to quantify the importance of this correlation. Next, using the theory of CTRWs we explain a to date unexplained phenomenon of why dispersion coefficients for a fixed Péclet number increase with increasing Reynolds number, but saturate above a certain value. Finally we also demonstrate that effective preasymptotic models that do not adequately account for velocity correlations will also not predict asymptotic dispersion coefficients correctly.