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  • Print publication year: 1998
  • Online publication date: October 2011

5 - The Theory of Unsteady Separation

Summary

The Analogy between Unsteady Separation and Separation from a Moving Surface

For the steady fluid motions considered in the preceding chapters, flow separation leads to a change in the flow structure as a whole. The limiting state when the Reynolds number tends to infinity is defined by the Helmholtz–Kirchhoff theory for ideal fluid flows with free streamlines, and the location of the separation point, in accordance with the criterion of Prandtl, coincides with the point where the shear stress at the surface of the body vanishes (see Chapter 1).

The situation is somewhat different if the flow is unsteady. To illustrate this fact we consider the example of flow past a circular cylinder that is set into motion instantaneously from rest. The starting motion of a body in a viscous fluid can be likened to the introduction of the no-slip condition at the surface of a body as it moves through a fluid that has no internal friction. Therefore, at the first instant the flow is potential and is described by the well-known solution for unseparated steady flow past a cylinder with zero circulation. At the body surface, where the no-slip condition is imposed, there arises the process of vorticity diffusion and convection, which leads to the formation of a boundary layer.

Blasius (1908) formulated this problem and gave an approximate solution. It turns out that at a certain instant the point of zero skin friction starts moving upstream along the body surface from the rear stagnation point. At the same time a region of reverse flow appears within the boundary layer (Figure 5.1).