Three-dimensional advective–diffusive boundary layers in open channels with parallel and inclined walls

Abstract

We study the steady laminar advective transport of a diffusive passive scalar released at the base of narrow three-dimensional longitudinal open channels with non-absorbing side walls and rectangular or truncated-wedge-shaped cross-sections. The scalar field in the advective–diffusive boundary layer at the base of the channels is fundamentally three-dimensional in the general case, owing to a three-dimensional velocity field and differing boundary conditions at the side walls. We utilise three-dimensional numerical simulations and asymptotic analysis to understand how this inherent three-dimensionality influences the advective–diffusive transport as described by the normalised average flux, the Sherwood Sh or Nusselt numbers for mass or heat transfer, respectively. We show that Sh is well approximated by an appropriately formulated two-dimensional calculation, even when the boundary layer structure is itself far from two-dimensional. This is a key and novel results which can significantly simplify the modelling of many laminar advection–diffusion scalar transfer problems. The different transport regimes found depend on the channel geometry and a characteristic Péclet number Pe based on the ratio of the cross-channel diffusion time and the longitudinal advection time. We develop asymptotic expressions for Sh in the various limiting regimes, which mainly depend on the confinement of the boundary layer in the lateral and base-normal directions. For Pe≫1 we recover the classical Lévêque solution with a cross-channel-averaged shear rate γ1/3¯,Sh∝γ1/3¯Pe1/3, for both geometries despite strongly curved boundary layers; for parallel walls a secondary regime with Sh∝Pe1/2 is found for Pe≪1. In the case of truncated wedge channels, further regimes are identified owing to curvature effects, which we capture through a curvature-rescaled Péclet number Peβ=β2Pe, with β the opening angle of the wedge. For Pe1/2≪β≪1, the Sherwood number appears to follow Sh∼β3/4Peβ1/16. In all cases, we offer a comparison between our three-dimensional simulations, the asymptotic results and our two-dimensional simplifications, and can thus quantify the error in the flux from the simplified calculations. Our findings are relevant to heat and mass transfer applications in confined U-shaped or V-shaped channels such as for the decontamination and cleaning of narrow gaps or transport processes in chemical or biological microfluidic devices.