Distribution of Singularities on the Surface of a Body

A distribution of vortex singularities is used to represent a thin airfoil in the devel­opment of an analytical thin-airfoil solution. A distribution of vortex singulari­ties also is used in the treatment of three-dimensional lifting wings in Chapter 6. Distributions of other types of singularities are useful for certain applications, which now are introduced. For convenience, a distribution of singularities running from Point 1 to Point 2 along a Cartesian coordinate axis is considered for simplicity. In this interval, 1-2, there is an infinite number of singularities of infinitesimal strength; such a distribution is called a vortex sheet.

It is demonstrated in Chapter 4 that there is a jump in tangential velocity across a vortex sheet—that is, Am ф 0. As a result, the circulation, Г, around the sheet between x1 and x2 (Fig. 5.6) is nonzero.

It follows that a vortex sheet is a useful representation for a lifting body. The vortex is the singularity chosen in this chapter to analyze the behavior of two­dimensional airfoils and in later chapters for three-dimensional wings and bodies of revolution at angle of attack.

A distribution of sources (and sinks) is useful for flows that are symmetrical about an axis as depicted in Fig. 5.7. This type of distribution of singularities is used to treat bodies of revolution at zero angle of attack in Chapter 7.

Across the source sheet, u is continuous; however, there is a jump in w across the x-axis. Thus, the source sheet splits the streamlines and represents body thickness (recall the superposition of sources and sinks described in Chapter 4). The circu­lation around the source sheet is zero so that it is not useful for representing a lifting body. However, the source singularity may be used in conjunction with a vortex distribution to represent the thickness effect on a lifting body with finite thickness.

A doublet distribution also may be used to represent a lifting body. This state­ment must be considered carefully because a doublet was superposed with a uni­form stream in Chapter 4 to represent the flow around a nonlifting cylinder and a vortex singularity was added to produce asymmetric flow and lift. The distinction is that the doublet used to generate the flow around the cylinder was developed by placing the source-sink pair along the x-axis (i. e., streamwise direction) and then considering the limiting case with the two singularities meeting at the origin. The resulting doublet then was said to have its axis in the x-direction.

z, w >

Figure 5.6. Vortex-singularity distribution.

Now consider a source-sink pair located on the z-axis. Let the source be at the origin of the coordinate system and the sink be located on the positive z-axis, as shown in Fig. 5.8(a). This represents the same source-sink pair in Chapter 4 rotated clock­wise by 90 degrees. The streamlines are as shown. Now, we generate a doublet at the origin of the coordinates by letting h ^ 0 while keeping the product (Ah) a constant, where Л is the strength of the source-sink pair (Fig. 5.8(b)). We focus our attention on the flow direction at the origin and let the doublet at the origin represent one of a series of doublets of infinitesimal and variable strength distributed along the x-axis (but with the doublet axes in the z-direction), as shown in Fig. 5.8(c). Notice that the resulting doublet-sheet flow in the z-direction is into the top of the sheet and out of the bottom. Also note that this is not the same behavior found in the source sheet.

Detailed analysis (see Karamcheti, 1980) shows that there is no jump in w across the sheet. However, there is a jump in the velocity potential across the sheet and, hence, a jump in the u component of velocity across the sheet. This means that the circulation around the doublet sheet is nonzero. It follows that a two – or three-dimen­sional lifting body in a freestream can be represented by superposing a uniform flow and a distribution of doublet singularities on the surface of the body with the axes of the doublets in the z-direction. Physically, the doublet sheet may be thought of as imparting a downward momentum to the oncoming flow similar to a vortex sheet. This change in z-momentum generates a net lift force on the sheet and, therefore, on the body it represents.

Thus, flows involving lifting bodies may be modeled using either a vortex or a doublet distribution. In fact, on analysis, the local vortex-sheet strength per unit length, у (x), can be identified with the derivative of the doublet-strength distribution per unit length, dX/dx. In this textbook, the vortex-sheet distribution is used to rep­resent a lifting body rather than the doublet distribution because the interaction of the vortex and the flow is physically more apparent. The concept of lift introduced in Chapter 4 by means of a cylinder with circulation induced by a bound vortex then can be adapted readily to the study of airfoils.