Boundary layer growth

Abstract

When relative motion of a viscous incompressible fluid of constant density and an immersed solid body is started impulsively from rest, the initial motion of the fluid is irrotational, without circulation. This is shown by observation, and may be seen in many of the published photographs of fluid flow. The theoretical proof is exactly the same as that given, for inviscid fluids, in treatises on hydrodynamics; for it may be assumed that the viscous stresses remain finite. The fluid in contact with the solid body is, however, at rest relative to the boundary, whilst the adjacent layer of fluid is slipping past the boundary with a velocity determined from the theory of the velocity potential. There is thus initially a surface of slip, or vortex sheet, in the fluid, coincident with the surface of the solid body. In other words, there is a “boundary layer” of zero thickness. The vorticity in the sheet diffuses from the boundary further into the fluid, and is convected by the stream. The boundary layer grows in thickness. (The same results follow from a consideration of the equations for the vorticity components in a viscous incompressible fluid, or of the equation for the circulation in a circuit moving with the fluid.)