8. Kern E. Kenyon, Curvature Boundary Layer

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Volume 16: Pages 74-85, 2003

Curvature Boundary Layer

Kern E. Kenyon

4632 North Lane, Del Mar, California 920144134 U.S.A.

The curvature boundary layer is defined to be the mechanical boundary layer that occurs when a steady homogeneous frictionless flow of arbitrary speed and semiinfinite extent passes by a solid, rigid, and impermeable boundary with curvature in it, and both fluid and boundary are at the same temperature. Assuming no flow separation, the fluid adjacent to the boundary must follow the curving surface, creating an acceleration of the flow normal to the streamlines. For steady motion the crossstream acceleration of the fluid particles is balanced by an equal but opposite pressure force, resulting in a pressure variation along the boundary that is sustained by the boundary's rigidity. Along the streamlines Bernoulli's law applies: the pressure is least where the speed is greatest and vice versa. These physical statements, plus conservation of mass for an incompressible fluid, provide the basis for calculating the thickness of the curvature boundary layer for boundaries with small average slopes. If the boundary configuration can be characterized by an amplitude and a wavelength, which are both arbitrary so long as the amplitudetowavelength ratio is small compared to one, then the result of the calculation is that the perturbations in the fluid motion and pressure caused by the boundary variation decay exponentially away from the boundary with an efolding scale proportional only to the wavelength of the boundary. The larger the wavelength, the farther the perturbations penetrate into the fluid. Examples worked out in some detail are for a thin circular arc wing and for a sinusoidal solid wall. Surface gravity waves are shown to be a special case of the curvature boundary layer when the mean flow speed past the sinusoidal wall has the one particular value (which equals the wave phase speed) that allows the solid wall to be removed and replaced with still air without disrupting the steady motion of the fluid. This is consistent with a littleknown thought experiment of Einstein's [Naturwissenschaften 4, 509 (1916)].

Keywords: inviscid boundary layer, curved boundary, centrifugal force, exponential decay

Received: August 19, 2002; Published online: December 15, 2008