3 - Self-similarities of the second kind: first examples

Summary

Flow of an ideal fluid past a wedge

3.1.1 The problem statement and a direct application of dimensional analysis

In the heat conduction and gas dynamic problems discussed in the preceding chapter, we were able to establish that the intermediate-asymptotic solutions were self-similar and to determine the self-similar variables using dimensional analysis alone. However, this is very frequently not the case. It often turns out that a problem has a self-similar intermediate-asymptotic solution, but that dimensional analysis alone is insufficient to obtain it. We shall presently see this for a simple example – the flow of an ideal fluid (one having no internal friction) past a wedge. Even this simple example will help us to find out why dimensional analysis on its own is sufficient for constructing self-similar solutions in some cases but not in others. Further examples will demonstrate other, more complicated situations. Thus, in this section, we shall consider the steady symmetric flow of an ideal fluid of constant density past a wedge-shaped body with uniform velocity U at infinity perpendicular to the leading edge of the wedge (Figure 3.1). We select a system of rectangular coordinates as shown in Figure 3.1 (a), with the x-axis in the plane of symmetry of the wedge, parallel to the flow velocity, the z-axis along the leading edge of the wedge, and the y-axis perpendicular to these two axes. It may be assumed that the motion is identical in all planes perpendicular to the leading edge of the wedge. Thus, the z-component of the velocity, u_z, can be set equal to zero, while the longitudinal and transverse components of the velocity,u_x and u_y, depend only on the coordinates x and y. The region in which the fluid moves is very large; we shall assume that it is infinite. It is also an important assumption; however, for the present case it is not critical.