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Abstract We derive a class of exact solutions for Stokes flow in infinite and semi‐infinite channel geometries with permeable walls. These simple, explicit, series expressions for both pressure and Stokes flow are valid for all permeability values. At the channel walls, we impose a no‐slip condition for the tangential fluid velocity and a condition based on Darcy's law for the normal fluid velocity. Fluid flow across the channel boundaries is driven by the pressure drop between the channel interior and exterior; we assume the exterior pressure to be constant. We show how the ground state is an exact solution in the infinite channel case. For the semi‐infinite channel domain, the ground‐state solutions approximate well the full exact solution in the bulk and we derive a method to improve their accuracy at the transverse wall. This study is motivated by the need to quantitatively understand the detailed fluid dynamics applicable in a variety of engineering applications including membrane‐based water purification, heat and mass transfer, and fuel cells.
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Exact solutions for ground effect
‘Ground effect’ refers to the enhanced performance enjoyed by fliers or swimmers operating close to the ground. We derive a number of exact solutions for this phenomenon, thereby elucidating the underlying physical mechanisms involved in ground effect. Unlike previous analytic studies, our solutions are not restricted to particular parameter regimes, such as ‘weak’ or ‘extreme’ ground effect, and do not even require thin aerofoil theory. Moreover, the solutions are valid for a hitherto intractable range of flow phenomena, including point vortices, uniform and straining flows, unsteady motions of the wing, and the Kutta condition. We model the ground effect as the potential flow past a wing inclined above a flat wall. The solution of the model requires two steps: firstly, a coordinate transformation between the physical domain and a concentric annulus; and secondly, the solution of the potential flow problem inside the annulus. We show that both steps can be solved by introducing a new special function which is straightforward to compute. Moreover, the ensuing solutions are simple to express and offer new insight into the mathematical structure of ground effect. In order to identify the missing physics in our potential flow model, we compare our solutions against new experimental data. The experiments show that boundary layer separation on the wing and wall occurs at small angles of attack, and we suggest ways in which our model could be extended to account for these effects.
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- PAR ID:
- 10189014
- Date Published:
- Journal Name:
- Journal of Fluid Mechanics
- Volume:
- 891
- ISSN:
- 0022-1120
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/jfm.2020.149
