Linearized boundary conditions are a commonplace numerical tool in any flow problems where the solid wall is nominally flat but the effects of small waviness or roughness are being investigated. Typical are stability problems in the presence of undulated walls or interfaces, and receptivity problems in aerodynamic transition prediction or turbulent flow control. However to properly pose such problems, solutions in two distinguished mathematical limits have to be considered: a flattening limit, where not only roughness height but also its aspect ratio becomes smaller and smaller, and a shrinking limit, where the size of roughness decreases but its aspect ratio need not. Here a connection between the two solutions is established through an analysis of their far-field behaviour. As a result, the effect of the surface in the shrinking limit, obtained from a numerical solution of the Stokes problem, can be recast as an equivalent flattening linearized boundary condition corrected by a suitable protrusion coefficient (related to the protrusion height used years ago in the study of riblets) and a proximity coefficient, accounting for the interference between multiple protrusions in a periodic array. Numerically computed plots and interpolation formulas of such correction coefficients are provided.

### Linearized no-slip boundary conditions at a rough surface

#### Abstract

Linearized boundary conditions are a commonplace numerical tool in any flow problems where the solid wall is nominally flat but the effects of small waviness or roughness are being investigated. Typical are stability problems in the presence of undulated walls or interfaces, and receptivity problems in aerodynamic transition prediction or turbulent flow control. However to properly pose such problems, solutions in two distinguished mathematical limits have to be considered: a flattening limit, where not only roughness height but also its aspect ratio becomes smaller and smaller, and a shrinking limit, where the size of roughness decreases but its aspect ratio need not. Here a connection between the two solutions is established through an analysis of their far-field behaviour. As a result, the effect of the surface in the shrinking limit, obtained from a numerical solution of the Stokes problem, can be recast as an equivalent flattening linearized boundary condition corrected by a suitable protrusion coefficient (related to the protrusion height used years ago in the study of riblets) and a proximity coefficient, accounting for the interference between multiple protrusions in a periodic array. Numerically computed plots and interpolation formulas of such correction coefficients are provided.
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Utilizza questo identificativo per citare o creare un link a questo documento: `http://hdl.handle.net/11386/4027924`
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